A History of t he Axi omatic Formulation of Probabili ty from Borel to Kolmogorov" Part I JACK BARONE & ALBERT NOVIKOFF Communicated by M. KLINE Abstract This paper, the first of two, traces the origins of the modern axiomatic formulation of Probability Theory, which was first given in definitive form by KOLMOGOROV in 1933. Even before that time, however, a sequence of develop- ments, initiated by a landmark paper of E. BOREL, were giving rise to problems, theorems, and reformulations that increasingly related probability to measure theory and, in particular, clarified the key role of countable additivity in Probability Theory. This paper describes the developments from BOREL'S work through F. HAUSDORFF'S. The major accomplishments of the period were BOREL's Zero- One Law (also known as the BOREL-CANTELLI Lemmas), his Strong Law of Large Numbers, and his Continued Fraction Theorem. What is new is a detailed analysis of BOREL's original proofs, from which we try to account for the roots (psychological as well as mathematical) of the many flaws and inadequacies in BOREL's reasoning. We also document the increasing realization of the link between the theories of measure and of probability in the period from G. FABER to F. HAUSDORFF. We indicate the misleading emphasis given to independence as a basic concept by BOREL and his equally unfortunate association of a HEINE-BOREL lemma with countable additivity. Also original is the (possible) genesis we propose for each of the two examples chosen by BOREL to exhibit his new theory; in each case we cite a now neglected precursor of BOREL, one of them surely known to BOREL, the other, probably so. The brief sketch of instances of the "CANTELLI" lemma before CANTELLI's publication is also original. We describe the interesting polemic between F. BERNSTEIN and BOREL concerning the Continued Fraction Theorem, which serves as a rare instance of a contemporary criticism of BOREL's reasoning. We also discuss HAUSDORFF'S proof of BOREL' S Strong Law (which seems to be the fir st valid proo f of the theor em along the lines sketched by BOREL). In retrospect, one may ask why problems of "geometric" (or "continuous") probability did not give rise to the (KOLMOGOROV) view of probability as a form of
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of Pr obabi l i ty f r om Bor el to Kolmogorov" Par t I
J A C K B A R O N E & A L B E R T N O V I K O F F
Communicated by M. KLINE
Abstract
This paper , the f i rs t o f two, traces the or ig ins of the m od ern ax iom at ic
formula t ion of P robabi l i ty Theory , which was f i r s t g iven in de f in i t ive form by
KOLMOGOROV in 1933. Ev en b efore tha t t ime, howe ver , a seq uenc e of develop -
ments , in i t i a t ed by a l an dm ark pape r of E . BOREL, were g iv ing r is e to prob lems ,theorems , and re formula t ions tha t inc reas ingly re la ted probabi l i ty to measure
theo ry and , in pa r t i cu la r , c l a r i fi ed the key ro le of countab le ad di t iv i ty in
P r oba b i l i t y T he o r y .
T h i s p a pe r de s cr i be s t he de ve l opme n t s f r om B O R E L ' S w or k t h r ough
F. HAUSDORFF'S. The major accomplishments of the per iod were BOREL's Zero-
On e L aw (also kno wn as the BOREL-CANTELLI Lem mas) , his St ro ng L aw of La rge
Nu mb ers , a nd h i s Con t inue d Fra c t ion The orem . W hat i s new is a de ta i l ed ana lys i s
of BOREL's or ig ina l p roofs , f rom w hich we t ry to acco unt for the root s
(ps yc ho log ica l a s we l l a s mathe mat ica l ) o f the man y f laws and inade quac ies in
BOREL's reasoning . We a l so document the inc reas ing rea l i za t ion of the l inkbe tween the theor ies of measu re and of prob abi l i ty in the pe r iod f ro m G . FABER to
F. HAUSDORFF. We indicate the mis leading emphasis given to independence as a
bas ic co nc ep t by BOREL and his equal ly un for tun ate assoc ia t ion of a HEINE-BOREL
lem ma wi th c oun tab le addi t iv i ty . Also or ig ina l is the (poss ib le ) genesi s we propo se
for each of the two examples cho sen by BOREL to exhib i t h i s new th eory ; in each
case we c it e a now neglec ted precurso r o f BOREL, one of them sure ly kno wn to
BOREL, the o the r , p rob ably so. The b r i e f ske tch of ins tances of the "CANTELLI"
lemma before CANTELLI 's publ icat ion is a lso or iginal .
W e descr ibe the interes t ing pole mic b etwe en F. BERNSTEIN an d BOREL
c onc e r n i ng t he C on t i nue d F r a c t i on T he o r e m , w h i c h se rve s as a r a r e i n st a nc e o f aco nt em po ra ry cr i t ic ism of BOREL's reasoning. W e also discuss HAUSDORFF'S pro of
of BOREL'S S t rong Law (which seems to be the f i rs t va l id pr oo f of the th eor em a long
the l ines sketched by BOREL).
I n r e t r o s pe ct , one ma y a s k w hy p r ob l e ms o f " ge o me t r i c " ( o r " c on t i nuo us " )
pro bab i l i ty did n ot give r ise to th e (KOLMOGOROV) view of pro ba bi l i ty as a form of
10. HAUSDORFF'S"Grundztige der Mengen lehre': A Notable Advance in Technique .. .. 182
10.1. General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18210.2. FIAuSDORFF'S Proo f of the St rong Law: The Use of Mome nt s a nd BIENAYMt~-
Th e overa l l purpose of th i s work is to ske tch the h ighpoin t s in the dev e lop me nt
o f t he a x i oma t i c f o r m u l a t i on o f P r oba b i l i ty T he o r y i n t e r ms o f me a s u r e t he o r y . (I ti s no t our in ten t ion to deprec ia te the reby o the r e f for t s to found the theory of
p r oba b i l i t y on a x i oma t i c ba s es t ha t do no t e mp l oy t he c on c e p t o f me a s u r e t he o r y. )
This axio ma t ic for m ula t io n was f i rs t def ini t ively achie ved by A. KOLMOGOROV in
1933 . T he ne e d f o r a n a x i oma t i c f ounda t i on f o r P r oba b i l i t y T he o r y ha d be e n
s t ressed by HILBERT as par t of the s ixth pr ob lem in his cele brate d l i st of 1900: " . . . :
To t rea t . .. b y means of ax ioms , those phys ica l sc iences (sic) i n w h ic h ma t he m a t i c s
p lays an impor tan t pa r t ; in the f i r s t rank a re the theor ies of P robabi l i ty and
M ech an ics" (HILBERT (1900: 81)) . Thu s this w ork can b e con s ide red a sketch of the
hi s tory of pa r t o f one of HILBERT's problems . Her e we t race the h i s to ry f rom the
wo rk of BOREL (1909) thr ou gh the w ork of HAUSDORFF (1914); subse que nt ly
Par t I I wi ll ca r ry the na r ra t ive forw ard to STEINHAUS, FRI~CHET, CANTELLI,
POLYA, MAZURKIEWICZ and others , culm inat in g in the wo rk of KOLMOGOROV.
The l and ma rk paper , in i t i a ting the m od ern theo ry of probabi l i ty , i s E . BOREL's
" L e s P r oba b il i ti 6 s D 6n omb r a b l e s e t L e u r s A pp l ic a t i ons A r i t hm6 t i que s " o f 1909 .
H e r e w e a r e ma i n l y c onc e r ne d w i t h t he c on t e n t s , ba c kg r ound , a nd i mme d i a t e
reac t ions to th is paper . The key f igures imm edia te ly fo l lowing BOREL are
a . FABER, F. BERNSTEIN,and F. HAUSDORFF. Th eir c on tr ib ut io ns wi l l be dis -
cussed in turn. Key predecessors are A.WIMAN (1900, 1901) , E.VANVLECK
(1908) and, in fact , BOREL himself by vi r tue of a br ie f no te of 1905. Bo th WIMAN
and VAN VLECK are subjects of separa te b r ief note s ; the no te o n VAN VLECK has
alr ead y ap pe are d (NOVIKOFF & BARONE (1977)). O f these earl ie r works, o nly
BOREC's paper of 1905 is discussed a t any length here .
An y a deq ua te d i scussion of the conten t s of BOREL'S l and m ark pap er (which we
shal l refer to as BOREL (1909)) i s of necess i ty del icate an d som ew hat deta i led. T he
reason for th i s is the i ron ica l c i rcum s tance tha t BOREL, the un que s t ioned foun der o f
me a s u r e t he o r y , a t t e mp t e d i n 1909 t o f ound a ne w t he o r y o f " de n um e r a b l e
p r o b a b i l i t y " without r e ly i ng on me a s u r e t he o r y . T he i r ony i s f u r t he r c omp oun de d
in the l igh t of BOREL's paper of 1905 which ident i fi ed "con t inu ou s p rob abi l i t y" in
the uni t in te rva l wi th measure theory the re . Consequent ly , we a re a t g rea t pa ins
bot h to es tab li sh and to co mp reh end BOREL'S re luc tance in 1909 to accept the
unde r ly ing ro le of counta b le addi t iv i ty in h i s new theory . The f ir s t pa r t o f th i s paper
is thus a n a t tem pt to e xam ine BOREL's s ta te of mi nd in 1909, taking into ac co un t his
ea r l i e r ins ight s and h i s re luc tance to explo i t them.
BOREL's paper , "Probab i l i t6s D6 nom brab les" , fal ls in to th ree m ajor d iv i s ions :
a "genera l t he ory " (cu lmina t ing in wha t we have ca l l ed the BOREL Ze ro- On e Law) ,
an appl i ca t ion of th i s theo ry to dec imal an d d yadic expan s ions ( the BOREL S t rong
L a w , o r S t r ong L a w o f L a r ge N um be r s ) , a nd a s e c ond a pp l i c a t i on o f t h is t he o r y ,
t h i s t i me t o C on t i nue d F r a c t i ons ( t he B O R E L C on t i nue d F r a c t i on T he o r e m) .
Since BOREL'S pa pe r is as inte restin g for i ts defects as for its results (as Pr ofe sso r
M. KAC once rem arked , " a l l o f i ts theorem s a re t rue bu t a lm os t a l l o f the proofs a re
fa lse") , we summarize i t s shortcomings in Sect ion 4.6.
BOREL'S immediate successors devoted themselves to c lar i fying both BOREL'S
S t r ong L a w a nd h i s C on t i nue d F r a c t i on T he o r e m by t r e a t i ng t he m bo t h w i t h i n
LEBESGUE'S the or y o f me asur e on the real l ine . 1 This c lar i f icat ion, howev er , was a t
the expense o f dras t i ca l ly d imini sh ing BOREL'S emph as i s o n the cons t ruc t io n o f a
ne w ge ne ra l t he o ry o f p roba b i l i t y c onc e rn i ng repeated independent trials.
STEINHAUS (1923) finally in co rp or at ed BOREL'S " g e n e r a l t he o ry" i n to me a s u ret he o ry by e xp l o i t i ng an a x i oma t i c c h a r a c t e r i z a t i on o f me a s u re t he o ry due t o
SIERPII~SKI (1918). STEINHAUS' w or k a nd m or e ge ner ally the in crea sing abs trac -
t ion of m eas ure the or y i t se l f ( ini t ia ted by FRt~CHET (1915) an d CARATHI~ODORY
(1914)) were events which helped pave the way to KOLMOGOROV's culminat ing
achievement . These deve lopments wi l l be d i scussed , among o the rs , in Par t
II.
Whi l e w e ha ve r e s t r a i ne d ou r s e l ve s f rom d ra w i ng ge ne ra l h i s t o r i og ra ph i c
conc lus ions , we be l i eve the conten t s of ou r pap er (both he re and in Par t I I ) furn i sh
am m un i t io n for those w ho wish to i l lus t ra te ( i) DIEUDONNI~ 'S " fu s ion " hy pothes i s
of mathematical progress (DIEUDONNI~ (1975:537)), (i i) LAKATOS' thesis on the
gradu a l r igo r iz ing of pa r t i a l ly un der s to od reason ing v ia d i spute (LAKATOS (1963-
64) ) , and ( i i i ) the doc t r ine tha t " spec ia l problems c rea te genera l theor ies" . The
present s tudy of BOREL'S wo rk br ings to l igh t a s tupen dou s ins tance o f tunne l
vis ion. I t a lso shows that a foundat ional ques t ion (such as HILBERT'S) may have
unex pec te dly comp lex responses. Th e so lu t ion w e t race to HILBERT'S proble m did
no t c om e f rom a d i r e c t a t t a c k ; i t e vo l ve d f rom c om pu t a t i ons w h i c h s uc ce s sfu ll y
deal t w i th a ser ies of par t ic ular , wel l -c hose n ques t ions , f i rs t ra ised by BOREL.
1. Brief Sketch of Major Results of Borel (1909)
BOREL cons id e red an inf in it e sequence of tr i al s , each hav ing only tw o
poss ib le outcomes a rb i t ra r i ly ca l l ed ' " success" and " fa i lure" . The probabi l i t i e s of
"succ ess" and " fa i lu re" on the n h t r i a l a re Pn and qn, respec tive ly , where
0_-<pn, qn<l,
p , + G = l , n = 1 ,2 , 3 , .. . .
Th e tr i a ls a re as sumed indep enden t . In con tem po ra r y t e rms such t ri a ls a re ca l led
"Binomia l " or "PoISSON" t r i a l s . In wha t represent s a dec i s ive new s tep , BOREL
asked for the p rob abi l i ty A k tha t exac t ly k successes occu r in such an inf in i t e
sequen ce (k = 0 , 1, 2 , . .. ) and , mo s t im por ta n t , the p roba bi l i ty A , of the occ ur ren ce
of inf in i t ely m an y successes.
There i s no nota t ion for the se t s (or "event s" ) under cons ide ra t ion in BOREL
(1909). In consequ ence , the re a re a l so no u nions , com plemen ts , in te rsec tions , etc.,
ind ica ted or re fe r red to as such .
1 T h i s s t a t e m e n t m u s t be qua l i f i e d by e m p ha s i z in g f ir s t t ha t N . WIE NE R a nd P . L ~ vY w or ke d a t a
different level, not follow ing BOREL'S lead directly, an d sec ond POLYA, CANTELLI an d MAZURKIEWlCZ
c ons ide r e d r a the r ge ne r a l " r a nd om va r i a b l e s " o r t r i a l s " a s i n B OR EL 'S "ge ne r a l t he o r y" , w i tho u t
BOREL sh ow ed tha t i f ~ p , co nve rge s , the n S = 1, whi le i f ~ p , diverges , S = 0. S ince1 1
A~o = 1 - S , t h e r e m a r k a b l e r e s u l t ( " B O R EL ' s Z e r o - O n e L a w " )
/°lf ~ p n c o n v e rg e s
A o 3 ~ 1oo
if ~ pn dive rges .1
is o b t a in e d . T h e r e st o f t h e p a p e r c o n t a i n s , a s i ts i m p o r t a n t r e su l ts , t w o a p p l i c a t i o n s
o f t h is c u r i o u s b e h a v i o r o f A ~ .
T h e f i rs t a p p l i c a t i o n c o n c e r n s t h e d y a d i c e x p a n s i o n o f a r e a l n u m b e r x c h o s e n
" a t r a n d o m " i n [ 0 , 1 ] :
x = . b l b 2 . . . b , . . . - ~ - b n1 2n
w h e r e b , = b,(x) i s e i t he r 0 o r 1 . I t is a s s um ed tha t t he se quen ce {b ,} is genera t ed , o r
t h e n u m b e r x is " c h o s e n " , s o t h a t e a c h b i n a r y d i g i t b , (x) h a s p r o b a b i l i t y ½ o f b e i n g 0
o r 1, a n d a l s o s o t h a t t h e v a r i o u s d i g it s n = 1, 2 , 3 . . . . a r e e x a m p l e s o f i n d e p e n d e n t
t r ia l s . BOREL cho ose s a f i xed sequen ce 2 , go ing t o i n f in i t y wi th n bu t t h a t
£ = 0i m ] / n "
L e t v, (x) d e n o t e t h e n u m b e r o f o n e s a m o n g t h e f ir st n b i n a r y d i g it s b~(x),b z (x ) , . . . , b n (x ) . F o r e a c h d y a d i c e x p a n s i o n B O R EL c o n s i d e r e d t h e a s s o c i a t e d
sequ ence o f " t r i a l s " t he n th t r ia l o f wh ich i s de f ined as success i f an d on ly i f
Ivz,(X)-nl~£~, n=1,2,3,....
L e t Pn b e t h e p r o b a b i l i t y o f t hi s o c c u r r e n c e a n d q , = 1 - p , t h e c o m p l e m e n t a r y
p r o b a b i l i t y . ( W e h a v e r e v e r s e d B O R E L 's n o t a t i o n f o r p , a n d q n t o c o n f o r m w i t h o u r
n o t a t i o n f o r Eoo.) B y a n a p p l i c a t i o n o f t h e C e n t r a l L i m i t T h e o r e m , B O RE L a t t e m p t sco
t o e s t ab l i s h t h a t ~ p , c o n v e r g e s , a n d h e n c e b y a n a p p l i c a t i o n o f h i s m a i n e a r l ie r1
r es u lt , t h a t Aoo = P ( E ~ ) = 0 . H e c o n c l u d e d t h e n t h a t t h e e ve n t,
h a s p r o b a b i l i t y 1 , i.e.,
lim Vzn(X) _ 1
,~oo 2n 2'
1 The formulas for A0, A 1,..., Ak, ... are extensions o f formulas for finite numbers of trials to thecase of denumerably many. The case of A~ is utterly new, since E~ is empty for a finite number of trialsand in that case its probability is of no interest.
This resu l t is ca lled var ious ly BOREL's Law of Large Num bers , the S t rong L aw of
Large Num bers , or BOREL's Law of No rm al Num bers . BOREL'S a rgu me nt wi ll be
anal yzed in § 5.
The second appl ica t ion cons iders the expan s ion in cont in ued f rac t ions of a" r a n do m l y" c hose n i r r a t iona l num be r c hose n f r om [ 0, 1 ]:
x = l
a l + l
a 2 + 1
a 3 + 1
+ 1
a . +i
° , .
Here each a , = a,(x) is a pos itive integ er, n = 1, 2, 3 .. ..
BOREL cons tructe d a sequence of tr ia ls , and associa ted "s ucce ss" an d
"fai lu re" , by ch oosin g a f ixed sequence ~b(n) and def ining for each ir ra t ion al x the n h
" t r ia l" as be ing a "success" or " fa i lure " (wi th cor responding probabi l i tie s p , , q , )
according as a ,>qb(n) or a , <¢(n) . Thus BOREL has assoc ia ted wi th each inf in ite
cont in ued f rac t ion an ins tance of an inf in i te sequence of tr ials , each t r ia l having
only two outcomes; he then seeks to apply h is Zero-One Law ment ioned above
to this collect ion of sequences of t r ia ls .By adro it m ani pu lat io ns of con tinu ed fract ions (see below), BOREL established
the inequali t ies
2 1 33 ( k + l ~ < P(a .(x)> k) < k +~"
cO
Repla cing k by q~(n) in this ineq uali ty sh ows tha t ~ p. converges if and only if
~ ~ j - does . 1Ag ain ap pealing to his earl ier result concerning A ~, BOREL asser ted1
tha t
1if ~ ( n ) converges, then with pro bab il i ty 1, a , wil l ul t im ately sa t isfy a , < qS(n)
whi le
1if ,~O~n) diverges, the n with prob abil i t y 1, a , wil l inf ini te ly of ten vio la te
tha t inequa l i ty .
This result w il l be cal led hencefo r th BOREL'S Continued Fraction Theorem.
2. "Countable Independence" as a Key Principle
As is evident f rom this br ief sketch, the proper ty of independence of tr ia ls
under l ies the formulas for Ao, A1, . . . ,A k, . . . and A~, and hence a lso the two
a pp l i c a t i ons . We p r opos e t o u s e t he ph r a s e "countable independence" for the
pr inc ip le tha t BOREL expl ic i t ly in t rod uce d a nd on which a l l o f h i s resu l t s a re based .
Thi s i s the as se r t ion , usua l ly t ak en as a hypothes i s , tha t a g iven co l l ec t ion of events ,
B1, B2 . . . . . Bn, . . . sa t is fy
P ( ~ B i ) = O P (B I). (2.1)
W hen the co l l ec t ion of event s is fin it e , the cor re spon ding pr inc ip le was know n as
the " lo i des probabi l i t6s com pos6es" , a l th oug h the no ta t io n for s e t in te rsec t ion was
not genera l ly emp loyed . BOREL assumed the pr inc ip le i f the event s B~ re fe r red to
d i f fe ren t t r ia l s for d i f fe ren t i and , m os t im por ta n t , a s sumed i t to h o ld even i f the
range of the index were inf in i te . A par t i cu la r ins tance o f spec ial impo r tance , which
we might ca ll the limited pr inc ip le o f countab le ind ependen ce , i s tha t (2 .1) ho lds i feach P(BI)= 1.T he r e a de r c a n s e e a n a na l ogy ( w h ic h w e be l ie ve mus t ha ve a c t e d pow e r f u l ly on
BOREL) wi th the b eh av ior o f length app l ied to d is joint intervals , B~, on the real l ine 1
1
This ex tens ion to the inf in i t e range and espec ia lly the in te rp re ta t io n o f the l e f t -handoo
s ide as a sor t o f genera l i zed l ength i f @ B i was no t i t se l f an in te rva l (or even
1express ible as a f ini te un ion of intervals ) l ies a t the hea rt of BOREL'S ear l ier ,
p r o f o und l y i mpo r t a n t d i s c ove r y o f t he t he o r y o f me a s u r e . I n pa r ti c u l ar , t he t he o r y
of measure , by focuss ing on th i s pr inc ip le ( "cou ntab le addi t iv i ty" ) i s l ed to e x tend
the c lass J o f in te rva l s to a mu ch wider and mo re s ign i f ican t cl as s N , wi th the key
pr op er ty tha t i f each of the Bi i s a se t of this w ider c lass N, and the B~ are d is joint ,oo
@ Bi is necessar i ly wi thin the c lass N.1
BOREL in 1909 may have fe lt h imse l f em bark ing on a s imi la r explora t ion , us ing
indep end ence o f t r ia l s a s the co unt e rp ar t to d i s jo in tness of in te rva l s and wi th
num e r i c a l p r o duc t s r e p l a c ing num e r i c a l s ums . I nde e d , t he ve r y na me B OR EL gavehi s theory , denumerable probab i l i ty , re fe rs prec i se ly to the ran ge of the index i in the
" lo i des probabi l i t~s com pos6es" . I t mos t ce r t a in ly does not r e fe r t o t he num be r o f
poss ib le d i s t ingui shable ou tcomes (i.e. sequen ces of t r ia ls ) as BOREL him self wel l
knew. (Fo r examp le , BOREL explo i t s the dy adic exp ans ion of num bers in the uni t
in te rva l a s an exam ple of h i s theory . The co l l ec t ion of such expan s ions has the
c a r d i na l num be r o f t he c on t i nuum. ) S om e how B OR EL f el t t ha t " de num e r a b l e
prob abi l i ty" , h is new theory , was poi sed be tween c las sica l " f in i t e pro bab i l i ty" an d
" ge o me t r i c p r oba b i l i t y " . G e om e t r i c p r oba b i l i ty , he w e ll kne w (cf BOREL (1905)),
ma y be ge ne r a li z e d to m a ke u s e o f h is ow n t he o r y o f me a s u r e a nd e ve n to m a ke u s e
o f L E B E S G U E ' S ne w i n t e g r a t i on t he o r y ( na me l y f o r t he c om pu t a t i o n o f me a nva lues ). In add i t ion to a " lo i des probab i l i t6s com pos6es" , c l as sica l p rob abi l i ty a l so
1 W h e n a u n i o n of disjoint se t s i s t a k e n , t h e sy m b o l @ A k wil l be used in s tead of U Ak t o e m p h a s i z e
the e vents B~ being assum ed dis joint (i.e. mut ua l l y i nc ompa t i b l e ) . I n s ome ke y
ins tances BOREL exte nd ed this a lso to th e inf ini te range,
co
P @ Bi) = ~ P(Bi) (countable addi t ivi ty) ,
but , as we shal l argue, a l l the internal evidence is that BOREL regarded countableindependenceas the essent ia l , new (and proba bi l is t ic) ingre dient o f his new theory .
By cont ras t he used countableadditivityse ldom (of ten sur rep t it ious ly) , and he never
explored i t s impl ica t ions ; about i t he had rese rva t ions so deep tha t he f requent ly
of fe red "a l t e rna t ive" proofs to evade re l i ance on i t .
This assessment , w hich we shal l defend by sui table analys is , explains a t leas t in
pa rt wh y BOREL fai led to dra w the co nclus ion , a t t r ib ute d to CANTELLI (1917a,
1917b), thatco
A c o = 0 i f ~ p , c o n v er g es1
even i f the t r ia ls are not a s s ume d i nde pe nde n t .
BOREL's fasc ina t ion wi th the pr inc ip le of counta b le indepe nden ce s imi lar ly
may expla in h i s fa i lu re to use anywhere n BOREL (1909) an arg um en t that
co
P(~Bi)<~P(Bi) (countab le sub-addi t iv i ty)
which fo l lows f rom coun tab le addi t iv i ty even i f the B~ a re not mut ua l l y i nc om-
pa t ib le . Wh ene ver an event is shown to hav e pro babi l i ty 0 , i t is no t b y pro of tha t i t
ha s " s ma l l c ove r s ", bu t r a t he r by p r oo f t ha t i t s c om pl e m e n t ha s p r ob a b i l i t y 1 . T h i s
l a t te r r e f o r m u l a t i on be c om e s a n a s s e rt i on a bo u t t he p r oba b i l i t y o f a n i n t e r se c t i on
and rest s on the pre fe r red pr inc ip le of coun tab le indep endence , o f t en in the
" l i m i t e d" f o r m.W e shal l sho w below (§ 7.2) tha t BOREL kn ew the CANTELLI pa rt of the BOREL-
CANTELLI lem ma s as ear ly as 1903, but in the co nte xt of a geom etr ic , not an
abs t rac t , space . M ore exac t ly , BOREL assoc ia ted th i s type of reasoning wi th the
HEINE-BOREL Cove r ing T he ore m as a pre l iminary . Thi s fur the r suppor t s ou r the -
s is tha t BOREL did n ot s ee the fu ll ana logy of prob abi l i ty w i th measu re except when
t h e p r o b l e m p e r m i t t e d a geometric interpretation.In our op in ion , the ana logy was
imper fec t ly seen even the n (cf the discuss ion of his exch ang e wi th F. BERNSTEIN,
§ 8).
3. Denumerable Probability versus Measure Theory
Befo re we exa min e BOREL's text in some deta i l , i t i s ins t ruct iv e to su m ma rize a
few more recent ly acqui red ins ight s . In contempora ry t e rms , each " t r i a l " cons id-
ered by BOREL is a 2-p oin t m easure space .Th e inf in it e sequences of ou tcom es a re jus t
poin t s of the (den ume rable ) CARTESIAN pro du c t of such 2-p oin t spaces. To each
sequence o f ou tc om es can b e as soc ia ted a sequence o f 0 's and l ' s, where fa i lu re
co rre spo nd s to 0, say, and success to 1. This pro vide s a m ap pin g fr om /~ {0, 1}~,
whic h is the CARTESIAN pr od uc t o f 2-p oint spaces , to the uni t interv al . This
ma p p i ng is 1 - 1 e xc e p t t ha t e a c h dya d i c r a t i ona l nu mb e r ma y ha ve t w o d i s t i nct
pre - images , one t e rmina t ing in 0 ' s , the o the r in l ' s . Fur the r , the as sumed
independence be tween d ig i t s un ique ly de te rmines the probabi l i ty , o r a s we would
say equiva len t ly now, measure , o f each dya dic in te rva l [Pn , P -~- I ] ; t h i s measur e ~
can be shown to sa t i s fy
m ( ~ B i )= ~ m(Bi)1
co
i f each B i is a d yad ic interv al , a nd @ B i i s again a dya dic in terval . F inal l y i t fol lows1
fro m al l this (by ins ights gaine d af te r 1909) that th e sequ ence {Pn} dete rm ines a a-
addi t ive measu re on the a -a lg ebra o f "BOREL se t s " N of [0 , 1 ]. I f p = l , n
= 1, 2 , 3 , . . . , t hen the m easure i s the ve ry one in t ro duc ed ea r l i e r by BOREL himsel f.
I f the {p,} a re no t iden t i ca l ly ½, the co r resp ond ing measu re i s a va r i an t o f the ab ove
(of the typ e la ter envis ioned by RADON (1913)) , a sor t of STIELTJES me asu re
assoc ia ted wi th a su i t ab le d i s t r ibu t ion func t ion F(x) bu t s t i ll def ined a t leas t on theBOREL se ts of [0, 1 ] . Thu s BOREL'S new th eor y o f" de nu m er ab le pro bab i l i ty" is, i f
each p , =½, es sent ia l ly a d i sgui sed ve rs ion of h i s own ea r l i e r theory of mea sure in
the uni t in te rva l , the proba bi l i s t i c im pac t of which h e rea l i zed a t l east pa r t i a l ly
w h e n t h e p r o b l e m w as o n e o f " g e o m e t r i c p r o b a b i l i t y " (cf § 7.3). There is no
evidence in favor of the sup pos i t ion tha t BOREL un der s too d th a t the case of genera l
{p ,} a l so gave r i s e to a "genera l i zed" BOREL measure (i.e., a d i f fe ren t countab ly
addi t ive m easure on the same cou ntab ly add i t ive f ie ld of sets). Th ere i s ev idence
t ha t i n t he c a s e p , =½ , n=1 , 2 , 3 , . . . , h e d i d s e ns e t he c onne c t i on be t w e e n
" p r oba b i l i t 6 s d dnom br a b l e s " a nd h is ow n t he o r y o f me a su r e . I n t hi s c a se he s pea ks
of the "p oin t de vue g6om 6t r ique" , re fe r r ing to the in te rva l [0 , 1 ] wi th i t s a s soc ia tedBOREL measure , and the "p oin t de vu e log ique" , re fe r r ing to the inf in i t e s equences
of t r ia ls , which w e deno te X {0, 1}i. BOREL asser ts that on e can em plo y e ither . (Cfi=1
§ 5.1 for the exact c i ta t ion. ) He m ake s this co m m en t o nly in discuss ing the specif ic
c a se p , =½ , n = l , 2 , 3 , . . . . O t he r i n t e r na l e v i de nc e t o s up por t a ll o f t he a bov e
asse r t ions concern ing wha t BOREL rea l i zed or rea l i zed imper fec t ly , wi l l be
di scussed be low.
Th e c ruc ia l ev idence aga ins t a s se r ting tha t BOREL rea l i zed tha t " de nu m era ble
p r o b a b i l i t y " was mea sure t heo ry in [0 , 1 ] is h is repea ted evas ion of counta b le
addi t iv i ty and cou ntab le sub -addi t iv i ty in a lm os t a l l o f h i s reasoning . Thi s ev idence
seems to us decis ive . A lesser evide nce of the same sort i s his seeing som ethi ng n ew
in the fac t tha t s e t s o f pro bab i l i ty ze ro ne ed no t be em pty . An d of course the re i s h i s
own as se r t ion ( in the face of h i s own con t ra ry as se r t ion for the case p ,=½, n
p r o d u c t o f 2 - p o i n t s p a ce s , is n o n - d e n u m e r a b l e , a s h a d b e e n s h o w n e a r l i er b y
G. C ANT OR . T h e r e p r e s e n t a t i o n o f t h e u n i t i n t e r v a l a s a p r o d u c t o f 2 - p o i n t s p a c e s is
a b r i l l i an t idea , bu t i t cann ot a l t e r i ts fami l ia r , and to BOP,EL r epug nant , ca rd ina l
n u m b e r c . T h e e v i d e n c e f o r t h e r e p u g n a n c e , a n d t h e m i s c o n c e p t i o n a s t o t h ec a r d i n a l n u m b e r o f t h e s a m p l e s p a c e s c o n s i d e r e d r e - e c h o e s i n th e c l o s i n g li n es o f
t h e p a p e r :
" A t s u c h t i m e a s t h e t h e o r y o f d e n u m e r a b l e p r o b a b i l i t i e s is d e v e l o p e d i n t h e
m a n n e r j u s t i n d i c a t e d , i t w il l b e i n t e r e st i n g t o c o m p a r e t h e r e s ul t s s o a c q u i r e d
wi t h t h o s e o b t a i n e d i n t h e t h e o r y o f c o n t i n u o u s o r g e o m e t r i c p r o b a b i l i t y .
I n t h e g e o m e t r i c c o n t i n u u m t h e r e exist cer ta inly ( if i t i s no t a misu se to
e m p l o y t h e v e r b to exist) s o m e e l e m e n t s wh i c h c a n n o t b e d e f i n e d : s u c h i s t h e
r e al s en s e o f t h e i m p o r t a n t a n d c e l e b r at e d p r o p o s i t i o n o f M r . G e o r g C a n t o r : t h e
c o n t i n u u m is n o t d e n u m e r a b l e . S h o u l d a d a y c o m e w h e n t h e s e undefinablee leme nts cou ld be pu t a s ide as no longe r neede d m ore or le s s impl ic i t ly , it wo uld
c e r t a i n ly b r i n g g r e a t s i m p l i f i c a t i o n in t h e m e t h o d s o f An a l y s i s ; I s h o u l d b e
h a p p y i f t h e p r e c e d i n g p a g e s c o u l d h e l p a r o u s e t h e i n t e r e st wh i c h t h e s t u d y o f
such ques t ions deserves ." ( I ta l ics in the or iginal . ) BOP,EL (1909: 271).
On e o f the b es t wa ys to fa i l to see a re la t io n (e.g . tha t X {0, 1}~ an d [0, 1] arei=~
equiva len t a s ca rd ina l s e t s and even as measure spaces ) i s to yea rn tha t no such
r e l a t i o n h o l d. B ORE L s u r e ly d i d n o t w i s h h i s o wn " d e n u m e r a b l e p r o b a b i l i t i e s " t o
jo in the con t inuum as a fu ture fos s i l , a mere t r ans i to ry dev ice .
4.2. The Calculation o f A o
BOREL turned f i r s t to e s tab l i sh ing the fo rmula fo r Ao
A 0 = (1 - p~)(1 - P2) -.. (1 - p, ) .. .. (4.1)
co
BOREL exc ludes in advan ce a ny p , = 1 so he can conc lude , i f~ . p , is conve rgent , tha t1
0 < A 0 < 1. ° ' I n the case of convergen ce , the ex tens ion of the p r inc ip le o f com po s i teprob abi l i t i e s goes wi tho ut s a y ing . . . " (BOREL (1909: 249). Fur t he r s ince the l imi t o f
the pa r t i a l p rod uc ts i s pos i t ive , they ap pr oa ch the i r l imi t wi th smal l r e la t ive e r ror a s
wel l a s abso l u te e r ror . I n conc lus ion , " . . . ; the pass age to the l imi t tha t w e have
pe r for me d thus does no t r a i se any d i f fi cu lt i es an d i s en t i r e ly jus t i fi ed . " BOREL
(1909: 249).
In f ac t , wha t BOREL i s sk imming over i s the l imi t r e la t ion
2 i m P ( @ B i ) = P ( @ B i ) . (4.2)
An a s s u m e d i n d e p e n d e n c e a s s u re s a d d i t i o n a l l y t h a t i f e a c h B i h a s p r o b a b i l i t y
T h e l i m i t r e l a t i o n (4 .2 ), h o w e v e r , h a s n o t h i n g t o d o w i t h i n d e p e n d e n c e ; i t is o n e o f
t h e m a n y c o n s e q u e n c e s , a n d e v e n e q u i v a l e n t fo r m s , o f c o u n t a b l e a d d i ti v i ty . T h e
l i m i t r e l a t i o n (4 .2 ) is, f o r B O R E L, b o t h t o o d e s i r a b l e t o b e f a l s e a n d t o o e v i d e n t t o
r e q u i r e d i s cu s s io n o r e l a b o r a t i o n . I n d e e d , s in c e h e n o w h e r e e m p l o y s a n o t a t i o n f o rt h e a l g e b r a o f s e ts o r e v e n f o r s e ts t h e m s e l v e s o r f o r s e t f u n c t i o n s , it w o u l d n o t h a v e
b e e n e a s y f o r h i m t o s t a t e e x p l i c i t l y . H a d h e e m p l o y e d t h e s y m b o l i s m P ( @ B i ) ,
p e r h a p s h e m i g h t h a v e b e e n d r i v e n t o q u e s t i o n t h e d o m a i n o f t h e s e t f u n c t i o n P ( - )
j u s t a s h e h a d e a r li e r q u e s t i o n e d a n d e x t e n d e d t h e d o m a i n o f " l e n g t h " f o r p o i n t- s e t s
in [0, 1] .
T h e d i v e r g e n t c a s e h er e , a n d l a te r , c al ls f o r sp e c i al " p r e c a u t i o n s " . F o r
" p r o b a b i l i t 6 d i s c o n t i n u e " ( i . e . , f i n i t e s a m p l e s p a c e s ) p r o b a b i l i t y z e r o m e a n s
i m p o s s i b i l i ty . F o r " p r o b a b i l i t 6 c o n t i n u e " t h i s is n o t a b l y f a ls e , a n d B O R E L r e f e rs t h e
r e a d e r t o h i s o w n p a p e r o f 1 90 5. I n t h i s p a p e r ( c f § 7 .3 ) he ha d def ined t h e g e o m e t r i c
p r o b a b i l i t y o f B O R E L s e ts o n [ 0 , 1 ] a s b e i n g t h e i r m e a s u r e . T h i s e x t e n d s it s fa m i l i a r
d e f i n i t io n . I n t h is w a y h e p r o v e s , f o r i n s t a n c e , t h a t t h e p r o b a b i l i t y f o r a n u m b e r
p i c k e d a t r a n d o m t o b e i r ra t i o n a l is z e r o , e v e n t h o u g h t h is o u t c o m e is n o t
i m p o s s i b l e . H e n o w s a y s i t i s t h e s a m e i n " p r o b a b i l i t 6 d d n o m b r a b l e " : i n t h e
d i v e r g e n t c a s e t h e f o r m u l a ( 4 . 1 ) f o r A 0 g i v e s t h e v a l u e z e r o , s i n c e t h e r e l a t i o n
n
l i m 1 ~ ( 1 - p i ) = A on ~ o 3 1
is e v i d e n t l y a c c e p t e d , d e s p i t e t h e f a c t t h a t t h e s e q u e n c e o f u n b r o k e n s u c c e s se s
h a p p e n s t o e x is t. N o n - e m p t y s ets m a y h a v e z e r o - " p r o b a b i li t6 d 6 n o m b r a b l e " j u s t a s
n o n - e m p t y s e t s ( f o r e x a m p l e , t h e r a t i o n a l s ) m a y h a v e m e a s u r e z e r o ; h e d o e s n o t
c l a i m t h a t t h e y a re s e ts o f m e a s u r e z e r o s i n c e h e i n t r o d u c e s n o m e a s u r e .
4.3 . The Calculation o f A1 , A2, and A k
T h e d e r i v a t io n o f t h e f o r m u l a f o r A 1 i n v o lv e s a m o r e o v e r t u s e o f c o u n t a b l e
a d d i t i v i t y , b u t , l i k e t h e h i d d e n u s e i n t h e l i m i t r e l a t i o n ( 4 . 2 ) , t h i s a p p l i c a t i o n
p a s s e s w i t h o u t c o m m e n t . I f t h e l o n e s u c c e s s i n a s e q u e n c e o f t r ia l s i s a t t h e n h
t r ia l , t h e n t h e p r o b a b i l i t y o f t h i s s e q u e n c e i sG
A o = p , F I (1 - p i ) ,co~ 1 - P n i = 1
i ~ - n
i n t e r p r e t e d a s z e r o i n t h e d i v e r g e n t c a s e. I t fo l lo w s , b y t a c i t u s e o f c o u n t a b l e
a d d i t i v i t y , t h a t A a i s th e s u m o f th e s e c o,. I n t r o d u c i n g t h e n o t a t i o n
A I = 0 . " ( I t a l i c s i n t h e o r i g i n a l . ) B O R E L ( 1 9 0 9 : 2 5 0 ) .
W h y is A 1 = 0 a c o n s e q u e n c e o f l ir a a , = 0 ? P r e s u m a b l y b e c a u s e
A 1 = l i m a , .n~oo
L e t u s i n t r o d u c e s o m e n o t a t i o n l a c k i n g i n B O R E L'S p a p e r : w e d e n o t e b y E z t h e s et
o f s e q u e n c e s h a v i n g e x a c t l y o n e s u cc e ss , a n d b y E ~ th e s e t o f s e q u e n c e s w i t h e x a c t l y
o n e s u c c e s s a m o n g t h e f i rs t n o u t c o m e s . T h e r e l a t i o n b e t w e e n t h e s e ts E I a n d t h e
s e t s E ] i s t h a t , a s s e t s ,
E 1 = l i m E l .
I t f o l l o w s , by countable additivity, t h a t
A 1 = P ( E 1 ) = l ir a P (E ] )= l im ~r ,. 1n~o3 n~oo
T h e f a c t t h a t B O R E L c o u l d i m a g i n e h e h a d f o u n d a n alternative a r g u m e n t t oo3
e s t a b li s h A 1 = ~ c o , i n t hi s f a s h i o n o n l y r e in f o r c es t h e i m p r e s s i o n t h a t h e n e i t h e r
f u ll y re c o r g n i z e d w h e n h e w a s e m p l o y i n g c o u n t a b l e a d d i t iv i t y n o r a p p r e c i a t e d i ts
p r i m a c y i n l i m i t r e l a t i o n s .
T h e c a s e s A 2 a n d t h e n A k a r e t r e a t e d s i m i l a r ly . B O R E L w r i t e s
Ak = Ao ~ uil ... ui~
a n d n o t
A~ = ~ (Ao u ~ , . . . u ~ ) .
T h i s g a v e ri se t o t h e s a m e m i s g i v i n g s a b o u t t h e d i v e r g e n t c a s e q u o t e d a b o v e i n
c o n n e c t i o n w i t h ( 4 . 3 ) a n d ( 4 . 4 ) .
4.4. The Calculation of Ao3
T h e d e r i v a t i o n o f A o3 f u r n i s h e s s t i ll m o r e e v i d e n c e o f B O R E L 'S r e l u c t a n c e t o r e l y
f u l ly o n c o u n t a b l e a d d i t i v i t y . F i r s t, i n t h e c o n v e r g e n t c a s e , e a c h A k is p o s it i v e , k
= 1 , 2 , 3 , . . . , a n d i f w e d e f i n e S b y
t h e n
S = A o + A 1 + A 2 + " ' + A k + ' " ,
S = A o ( 1 + Z u f l + Z u i ~ u i 2 + . . . + Z u h u i 2 . . , u i + . . . )
= A o ( 1 + u l )( 1 + u 2 ) . . . (1 + u , ) . . . .
1 Theorem s evaluating lira P(E,)under v arious circumstances were the m ain results in the periodn~co
before BOREL 1909). BERNOULLI'S "W eak") Law of Large N umb ers, an d the Central Limit Theoremare theorem s of this type. They did n ot assert th at this lim it was itself the probab ility o f any single event,howe ver, wh ereas BOREL'S heorem did.On this im portant point, see the discussion in § 5.3 of BOREL'S("Strong") L aw o f Large Num bers.
L e t u s i n t r o d u c e t h e f o l l o w i n g n o t a t i o n :
E , , k = s e t o f t ri a l s eq u e n c e s f o r w h i c h t h e n th o u t c o m e o c c u r s e x a c t l y k t i m e s ,
E , , ~o = s e t o f tr i a l s e q u e n c e s f o r w h i c h t h e n th o u t c o m e o c c u r s i n f i n i te l y o f t e n ,
E , ,k = s e t o f t r i a l s e q u e n c e s f o r w h i c h t h e n th o u t c o m e o c c u r s a t m o s t k ti m e s .
I f w e c o n s i d e r a n y o n e i n d i v i d u a l o u t c o m e , s a y t h e n th, i t f o l lo w s f r o m t h e
" c o n v e r g e n c e " h y p o t h e s i s t h a t
P ( E n , ~ ) = 0
o r , e q u i v a l e n t l y ,
P ( G , o o )= 1.
N o w w e e n c o u n t e r a n e x a m p l e o f w h a t w e h a v e c al le d " l i m i t e d " c o u n t a b l e
i n d e p e n d e n c e ; a l t h o u g h t h e d if f er e n t o u t c o m e s n e e d n o t i n g e n e r a l b e i n d e p e n d e n t ,
t he eve n t s E c co,, n = 1 , 2 , . . . , h a v i n g p r o b a b i l i t y 1, a r e o f n e c e s si ty m u t u a l l y
i n d e p e n d e n t . I n w o r d s , g i v e n th e " f u l l y c o n v e r g e n t " h y p o t h e s i s , th e r e is p r o b a b i l i t y
1 t h a t a ll o u t c o m e s o c c u r o n l y f in i te l y o ft e n .
I n o u r n o t a t i o n , P ( U , , oo) = 1 fo r n = 1 , 2 , 3 , . . . im pl i es
P E c, oo = I ~ P ( E ~ , ~ ) = 1 . (4 .5 )1
T h i s l i n e o f r e a s o n i n g p r o v o k e s B O R E L t o o n e o f t h e r a r e i n s t a n c e s i n w h i c h
c o u n t a b l e i n d e p e n d e n c e ( e ve n in t h e " l i m i t e d " f o r m ) g iv e s r is e to e x p l ic i t d o u b t s :
I t m a y h e l p t o e x a m i n e t h e q u e s t i o n m o r e c lo s el y, t o b e s u re t h a t o u rr e a s o n i n g i s s t r ic t : t h e f a c t t h a t t h e r e i s a d e n u m e r a b l e i n f in i t y o f f a c t o r s e q u a l t o
1 c o u l d i n d e e d l e a v e s o m e d o u b t a s t o t h e v a l u e o f t h e i r p r o d u c t . B O R EL ( 1 9 09 '
255).
B O R EL t h e n r e a s s u re s h i m s e l f b y p r o v i d i n g a n e l a b o r a t e p r o o f o f a n a s s e rt i o n, i n
t w o p a r t s , w h i c h r e f i n e s a s w e l l a s " r e - e s t a b l i s h e s " t h e r e s u l t i n q u e s t i o n .
F i rs t , h e p r o v i d e s a n a r g u m e n t t h a t f o r e v e r y p o si ti v e i n te g e r k ,
( I n w o r d s , t h e p r o b a b i l i t y t h a t a l l o u t c o m e s o c c u r a t m o s t k t i m e s f o r a n y f i x e d f in i te
i n t e g e r k i s z e r o . ) T h i s i s e s t a b l i s h e d b y a n i n g e n i o u s c a l c u l a t i o n , t h e d e t a i l s o f
w h i c h a r e l e ft t o t h e r e a d e r ( c f B A R O N E (1 9 7 4 : 1 0 1 - 1 1 3 ) f o r a d e t a i l e d d i s c u s s i o n ) ,
t o g e t h e r w i t h t h e t a c i t l y a s s u m e d r e l a t i o n
P & , k = P ( & , O .n 1 n= l
T h i s r e l a ti o n i s t a n t a m o u n t t o a s s u m i n g i n d e p e n d e n c e o f t h e e v e n t s/ ~ , ,k , i . e . , o f
o u t c o m e s , n o t j u s t o f t ri a ls .S e c o n d , g i v e n a n y e > 0 , o n e c a n c h o o s e t h e s e q u e n c e k~ s o t h a t
s o p h i s t i c a te d r e f o r m u l a t i o n s o f c o u n t a b l e a d d i t i v i t y s u c h a s
P ( l i m E , ) = l i ra P ( E . ) ;
h e b a s e s h i s r es u lt s, s o m e t i m e s n e e d l es s ly , o n c o u n t a b l e i n d e p e n d e n c e a n d a s s u m e s
t h e m v a l i d i n c i r c u m s t a n c e s w h e r e n o i n d e p e n d e n c e h a s b e e n a s s u m e d o r
e s t a b li s h e d . C o u n t a b l e i n d e p e n d e n c e i t s e lf is t a k e n a s b e in g e v i d e n t b y a n a l o g y
w i t h t h e f i n i t e c a s e d e s p i t e o c c a s i o n a l r e s e r v a t i o n s . N o n o t a t i o n i s i n t r o d u c e d f o r
s et s, a n d h e n c e n o n e f o r se t fu n c t i o n s . T h e a d j e ct i ve " d e n u m e r a b l e " , a p p r o p r i a t e
f o r th e n u m b e r o f t ri a ls , is o c c a s i o n a l l y a n d e r r o n e o u s l y c o n s t r u e d t o r e fe r to t h e
s iz e o f t h e s a m p l e s p a c e , a n d i t i s as s e r te d t h a t t h e t h e o r y o f " d e n u m e r a b l e
p r o b a b i l i t i e s " is a m o r e " e f f e c t iv e " t h e o r y t h a n t h a t o f t h e c o n t i n u u m , s i n ce th e
l a t te r c l a im s t o t r e a t o f m o r e t h a n d e n u m e r a b l y m a n y e l e m e n t s a s b e i n g o n e
c o l l e c t i o n . ( C f t h e o p e n i n g r e m a r k a n d c o n c l u d i n g l i n es o f BO R EL 's p ap e r , b o t hg iven in § 4 .1 .) Th i s l a s t d i s t in c t io n i s o f cours e i l l u so ry .
H a d t h e p a p e r c o n t a i n e d n o m o r e , it w o u l d h a v e f u r n i s h e d n o e v i d e n c e
w h a t e v e r t o f a v o r t h e h y p o t h e s i s t h a t B O R E L u n d e r s t o o d t h a t h i s " p r o b a b i l i t 6 s
d 6 n o m b r a b l e s " w a s a f o r m o f m e a s u r e t h e o r y ( in p a r t i c u l a r t h a t c o u n t a b l e
a d d i t i v i t y w a s a n i n t ri n s ic p a r t o f i ts a p p a r a t u s ), a n d a ll t h e a f o r e m e n t i o n e d t o
d i s p u t e it . H o w e v e r , t h e t w o r e m a i n i n g s e c t i o ns o f t h e p a p e r s o m e w h a t c o u n t e r b a l -
a n c e t h i s o v e r - s i m p l e i n t e r p r e t a t i o n .
5. Borel's Chapter II: The Strong Law of Large N umbers
5 .1 . The Se t t ing o f the Prob lem
oO
C o n s i d e r t h e d e c i m a l e x p a n s i o n ~ ( b n / 1 0 " ), e a c h b~ b e i n g o n e o f t h e d i g i t s
0 , 1 , . . . , 9 . T h e m o r e g e n e r a l p r o b l e m o f " q - a r y " e x p a n s i o n s , ( b f f q ~ ) ,w h e r e e a c hn=l \
b , is a m o n g t h e i n t e g e r s 0, 1 , . . . , q - 1 , c a n o f c o u r s e b e t r e a t e d i n l i k e m a n n e r . )/
B O RE L p r o p o s e d t o s t u d y t h e p r o b a b i l i t y t h a t s u c h b n " b e l o n g t o a g i v e n s e t "
a s s u m i n g ( 1 ) t h e d i g i t s a r e i n d e p e n d e n t a n d , ( 2 ) e a c h d i g i t h a s e q u a l p r o b a b i l i t y(na m e ly 1 /10 ) o f a ch ie v ing the va lues 0 , 1, . . . , 9 .
I t is n o t n e c e s s a r y to e m p h a s i z e t h e p a r t l y a r b i t r a r y n a t u r e o f t h e s e tw o
h y p o t h e s e s : t h e f i rs t, i n p a r t i c u l a r , i s n e c e s s a r i l y i n a c c u r a t e w h e n o n e c o n s i d e r s ,
as one i f f o r ce d to in prac t i ce , t h a t a d e c i m a l e x p a n s i o n i s d e f i n e d b y a law,
w h a t e v e r m i g h t b e t h e n a t u r e o f t h a t l aw . I t m a y n o n e t h e l e s s b e in t e r e s t in g t o
s t u d y t h e c o n s e q u e n c e s o f t h i s h y p o t h e s i s , p r e c is e l y i n o r d e r t o r e a l iz e t h e e x t e n t
t o w h i c h t h i n g s o c c u r as i f t h i s h y p o t h e s i s w e r e v e r if ie d . T h e s e c o n d h y p o t h e s i s ,
t h a t i s t h e e q u a l i t y o f p r o b a b i l i t i e s f o r t h e v a r i o u s p o s s i b l e v a l u e s o f e a c h d e c i -
m a l d i g i t , s e e m s r a t h e r n a t u r a l , g r a n t i n g t h e f i r s t .
T h e s e t w o h y p o t h e s e s a r e e a s i ly j u s t if i e d a d d i t i o n a l l y b y t a k i n g n o t t h e
l o g i c a l, b u t t h e g e o m e t r i c p o i n t o f v i e w : t h e y a r e, i n d e e d , e q u i v a l e n t t o t h e
f o l l o w i n g : t he dec ima l number be ing r epre sen ted by a po in t o f the in t e rva l [0, 1],
the probabi l i ty that i t i s located in a subinterval i s equal to the length o f that
subinterval . O n e c o u l d i n t e r p r e t a n d v e r if y t h e r e s u lt s w e a r e g o i n g t o o b t a i n
f r o m t h is g e o m e t r i c p o i n t o f v i e w ; I w il l n o t d o s o , p r e f e r r i n g t o l e a v e a s i d e f o r
t h e p r e s e n t t h e t h e o r y o f c o n t i n u o u s p r o b a b i l i t y , w h i c h i s c o n n e c t e d , a s I h a v e
s h o w n e l s e w h e r e , t o t h e t h e o r y o f m e a s u r e o f s et s. 1 B O R E L (1 9 0 9 : 2 58 ).
T h u s B O R EL 'S " c o n s t r u c t i v i s m " l e a d s h i m t o a s s e r t t h a t t h e f ir st h y p o t h e s i s i s
" n 6 c e s s a i r e m e n t i n e x a c t e ' . O n e is l e d t o s p e c u l a t e t h a t B O RE L k n e w w h e r e h e
w a n t e d t o g o a n d a r r a n g e d t o g e t t h e re , a t t h e e x p e n s e o f hi s s c ru p l e s i f n e c e s s ar y .
T h e r e m a r k a b o u t t h e s e c o n d h y p o t h e s i s i s , p e r h a p s , e v e n m o r e i n t e r e s t i n g :
n o t h i n g i n t h e f i r s t c h a p t e r ( " p o i n t d e v u e l o g i q u e " ) d e m a n d s e q u a l p r o b a b i l i t i e s
1/q f o r e a c h b n. T h e g e n e r a l t h e o r y o f tr ia ls , e a c h w i t h q o u t c o m e s , a c c e p t s a n y
c h o i ce s p . . .. n = 0 , 1 . . . . q - l , s u c h t h a t
p o , s + p l , s + . . . + p q _ l , s = l , s = 1 , 2 , 3 , . . . .
5 .2 . The Spec ia l Case p , = 1 fo r Dyad ic Exp ans ions
B O R E L r e a l iz e s e x p l i c i tl y t h a t t h e s e c o n d h y p o t h e s i s o f e q u a l l y l i k e ly b , p e r m i t s
p r o b a b i l i t y t o b e i n t e r p r e t e d t w o e q u i v a l e n t w a y s , o n e o f w h i c h i s f a m i li a r. I n t h e
c a s e q = 2 , t h is m e a n s t h e p r o b a b i l i t y o f t h e d i g i t 0 a n d o f t h e d i g i t i i n t h e n t h p l a c e
a r e n o t o n l y i n d e p e n d e n t o f t h e c h o i c e o f di g it s in t h e o t h e r p l a c e s b u t a r e b o t h 1 . ( I t
is a n i n t e r e s ti n g c o m m e n t o n t h e c h a r a c t e r o f m a t h e m a t i c a l e v o l u t i o n t h a t b e f o r e
S TE IN H A U S ( 19 23 ) n o o n e h a d t h e t e m e r i t y to c o n s i d e r w h e t h e r a n y o t h e r c h o i c e o fp r e - a s s i g n e d p r o b a b i l i t i e s { P n} d e p e n d i n g i n g e n e r a l o n n c o u l d a l s o i n d u c e a
m e a s u r e o n [ 0 , 1 ] . 2) B O R E L n o w s t u d i e d t h i s e q u i p r o b a b l e c a s e b y h i s n e w
p r o b a b i l i s t i c m e t h o d s i n p r e f e r e n c e t o m e a s u r e - t h e o r e t i c o n e s ( al so h is o w n
i n v e n t i o n ) .
P r o c e e d i n g t o t h e c a s e o f o n l y t w o o u t c o m e s f o r e a c h t ri a l, t h e c a s e q = 2 , w i t h
t h e c o n v e n t i o n t h a t t h e d i g i t 0 i s a s u c c e s s , o r f a v o r a b l e c a s e , B O R E L s t a t e s
O n e k n o w s t h a t , if o n e c o n s i d e r s 2 n t ri al s, t h e p r o b a b i l i t y t h a t t h e n u m b e r o f
f a v o r a b l e c a s e s w i l l l i e b e t w e e n
is eq ua l to 0( ,~) , l e t t in g
n - , ; L ~ n a n d n ÷ ) d f f n
2 2
BOREL (1909: 259) .
1 T h i s i s th e s o l e r e f e r e n c e t o t h e t h e o r y o f m e a s u r e i n t h e e n t i r e p a p e r o f 1 90 9. I t s r o l e i s t o n o t i f y t h e
r e a d e r t h a t t h e t h e o r y o f m e a s u r e , a n a l t e r n a t i v e a p p r o a c h , i s no t b e i n g e m p l o y e d . B O R E L c l e a r l y i s
r e f e r r i n g t o h i s p a p e r o f 1 9 0 5 t h e c o n t e n t s o f w h i c h w i ll b e d i s c u s s e d i n § 7 .3 .
2 T h e s t u d y o f t h e s e m e a s u r e s , i m p l i c i t ly i n t r o d u c e d b y B O R E L, w a s f i r s t e x p l o r e d i n a n y d e t a i l a s
r e c e n t l y a s 1 9 4 7 (WINTNER) n d 1 9 4 8 (H A R TM A N ). T h e n o t i o n o f g e n e r a l i z i n g B O R R ( o r L EB ES GU E)
m e a s u r e t o o t h e r c o u n t a b l y a d d i t iv e s e t fu n c t i o n s h a d o c c u r r e d w i t h i n i n t e g r a t io n t h e o r y i n 1 91 3
( R A D 6 N ) a n d w i t h i n m e a s u r e t h e o r y i n 1 9 1 4 ( CA R ~r H~ O DO a Y) .
( T h e s t a t e m e n t i s o f c o u r s e u n t r u e , b u t t h e l imit o f t h i s p r o b a b i l i t y , a s n t e n d s t o
i n f i n i t y , i s g i ven by 0 ( 2 ) . ) As s um i ng
A n
l i ra ~nn = 0n ~ o o
b u t 2 , i t s e l f g r o w s u n b o u n d e d l y , B O R E L c o n s i d e r s e a c h s e t o f 2 n i n i ti a l tr i a ls a s
d e t e r m i n i n g a n e v e n t a s f o ll o w s . L e t v,(x) b e t h e n u m b e r o f z e r o s i n t h e f i rs t n d i g it s
o f t h e b i n a r y e x p a n s i o n o f x . F o r e a c h n , t h e " t r i a l " e x a m i n e s v2,(x) , a n d t h e r e s u l t is
a " s u c c e s s " i f
I v z , ( X ) - n l > 2 , 1 f n
a n d a " f a i l u r e " i f
Ivz,(x)-nl <& I/L
T h e p r o b a b i l i t y p , o f a f a v o r a b l e c as e is a ss e rt e d t o b e
2 ~ e_~2d2
a n d t h e p r o b a b i l i t y q , o f a n u n f a v o r a b l e c a s e is g i ve n b y q , = 1 - p ,. 1
B O R E L n o w focuses h is a t t e n t i o n o n t h e s et E ~ o f t h o s e " d y a d i c e x p a n s i o n s " f o rw h i c h i n f in i t el y m a n y " t r i a l s " h a v e " s u c c e s s e s " . A l w a y s f a s c in a t e d b y s e ts o f r e a l
n u m b e r s c h a r a c t e r iz e d b y a d e n u m e r a b l e s et o f c o n d i ti o n s , B OR EL h a d ju s t f o u n d an
n e w s e t, w i th a n e w " d e n u m e r a b l e " d e s c r ip t io n . F u r t h e r m o r e , s i nc e ~ , p , c o n v e r g e s1
i f 2 , g r o w s s u f f i c i e n t l y f a s t (e .g . n~ ), h i s Z e r o - O n e L a w a p p l i e d t o t h is c a s e ( a s s u m i n g
t h e v a l id i t y o f i ts a p p l i c a t io n ) a s s e r ts t h a t A ~ = 0 a n d s o P ( E ~ ) = 1 .
1 This ca lculation of p. and q. is serious ly flawed. Wha t is true is that p. and
oO
2 ~ e_a~ d2
are of the same o rder, b ut since ~.. is not fixed, this is not a case of the classical Central Limit Theorem.
The information needed, that
2 ~o
is a refined and relatively recent result. Without it the convergence of
(which is true if2. grows rapidly enough, e.g., n+) need not imply the convergence of~, p.. However, even
assumin g }~p. converges, one cannot conclude f rom BOREL'S result for A~ that there is a zero
probability of infinitely ma ny un favorable cases, since that result was established on the hypothesis that
the separate trials be independent. The various "trial s", with probability p. a nd q. o f success and failure
B O R E L a s se r ts ( w i t h o u t a r g u m e n t ) t h a t b o t h t y p es o f n u m b e r s a r e o f p r o b a b i l i t y 1.
W h a t is b e i n g t a c i t l y a f f i r m e d i s, a s u s u a l, t h a t
a n d t h a t
P = P ( N q ~ )k= l
q=2
w h e r e e a c h f a c t o r o n t h e r i g h t - h a n d s i d e is 1. T h e s e a r e c a se s o f " l i m i t e d " c o u n t a b l e
i n d e p e n d e n c e . A l a c o n i c f o o t n o t e r e a ds
I d o n o t t h i n k i t s e rv e s a n y p u r p o s e t o r e p e a t t h e d e t a i le d p r o o f o f t h e f a c t
t h a t o n e h a s t h e ri g h t t o a p p l y t h e t h e o r e m o f c o m p o s i t e p r o b a b i l it i e s , d e s p i t et h e d e n u m e r a b l e i n f in i ty o f c as e s. B O R g L ( 1 9 0 9: 2 6 1 : F o o t n o t e (5)).
O f c o ur s e, n o " d e t a i l e d p r o o f o f t h e r ig h t t o a p p l y t h e t h e o r e m o f c o m p o u n d
p r o b a b i l i t i e s " i n t h e d e n u m e r a b l y i n fi n it e c a s e h a s b e e n g i v e n ; r e ca l l th a t i t w a s fi rs t
m e n t i o n e d i n c o n n e c t i o n w i t h
A 0 = ( 1 - p l ) ( 1 - P 2 ) . . . (1 - p , ) . . .
" D a n s le c a s d e la c o n v e r g e n c e , l ' e x t e n s i o n d u p r i n c i p e d e s p ro b a b i l i t 6 s
c o m p o s 6 e s v a d e s o i, . . . " B O R E L (1 9 0 9 : 2 49 ). R e p e a t e d u s e h a s b y t h i s p o i n t
t r a n s f o r m e d t h e p r in c i p l e f r o m a s e lf - e v id e n t t r u t h t o o n e w h o s e p r o o f h a s b e e nd e m o n s t r a t e d a ll t o o o f te n .1
B O R E L ' s r e a s o n i n g , u p t o t h i s p o i n t , i s c h i e f ly f l a w e d i n th e f o l l o w i n g t w o w a y s :
h e e m p l o y s a r e su l t b a s e d o n i n d e p e n d e n c e o f e v e n t s f o r d e p e n d e n t t ri a ls ( th is f la w
w o u l d b e r e m e d i e d b y a " C A N T E L L I " m o d i f i c a t i o n o f h is Z e r o - O n e L a w , c f § 6.4)
a n d h e u s es a f o r m o f t h e C e n t r a l L i m i t T h e o r e m m o r e p r e c is e t h a n w a s t h e n
a v a il a b le . H A U S D O R F F ' s a n d l a t e r p r o o f s w e r e t o a v o i d a n y a p p e a l t o t h e C e n t r a l
L i m i t T h e o r e m w h a t e v e r .
A n a d d i t i o n a l o m i s s i o n is n o t e w o r t h y : d e s pi te a c h o i c e o f n o t a t i o n i d e nt i ca l t o
t h e o n e h e h a d e m p l o y e d f o r d is c u s si n g t h e (B E R N O U LL I) L a w o f L a r g e N u m b e r s i n
h i s t e x t - b o o k o n P r o b a b i l i t y , w r i t t e n e a r li e r i n t h e s a m e y e a r ( BO R E L ( 1 90 9 a : 6 3 -65)), h e m a k e s n o a t t e m p t t o c o m p a r e t h e B E R N O U L L I L a w a n d t h e n e w r es u lt . A
g r e a t o p p o r t u n i t y is l o st t h e r e b y : t r e a t i n g b o t h w i t h in t h e t h e o r y o f m e a s u r e , o n e
w o u l d h a v e b e e n le d t o th e c o m p a r i s o n o f c o n v e r g e n c e " i n m e a s u r e " a n d
c o n v e r g e n c e " a l m o s t e v e r y w h e r e " , a n t i c i p a t i n g t h e t r e a t m e n t s o f S LU T SK Y ( 19 25 ),
F RI~ CH ET (1 9 3 0) a n d t h e e a r l i e r s u c h c o m p a r i s o n s b y C A N T E L L I ( 1 91 7 ) a n d P O L Y A
( 19 21 ). I t is p o s si b l e t h a t t h e s t r o n g e r c h a r a c t e r o f h i s r e su l t b y c o m p a r i s o n w i t h
1 In discussing A0, Ak, A~ independence of trials was assumed. The independence o f the eventsNq, q = 2, 3,..., or o f he events Nqk, k = 1, 2, 3,..., is slightly mo re sophisticated. It is no t to be regarded asan additional ad hoc assumption. Rather one must establish the general fact that if P ( N ) = 1, then
P ( N c ~ A ) = P ( N ) P ( A ) . i .e ., N is independent of a n y other eve nt A. This BORE5 failed to d o.The proo f involves considering the complement of N c~A and then concluding that a set contained
in a set of probability zero m ust itself have probability zero. The form er consideration has the p ossibledrawback, for BOREL,of shifting attention away from "independenc e". As to the latter, BOREL s knownto have rejected the corresponding reasoning w hen employed in the context o f measure theory.
V LE CK h a d m a d e B O RE L's a c q u a i n t a n c e i n F r a n c e i n N o v e m b e r 1 9 0 5 - J a n u a r y
1 90 6, w h i l e o n s a b b a t i c a l l e av e , a n d t h e t w o r e m a i n e d o n f r ie n d l y t e r m s f o r y e a r s
t h e r e a f t e r . H a d h e c o m m u n i c a t e d h i s r e s u l t s ( i n c l u d i n g h i s c o n j e c t u r e a b o u t t h e
m e a s u r e o f V 0) t o B O R E L a t o n c e , i t is p o s s i b l e t h a t B O R E L w o u l d h a v e s e e n t h a t t h e
q u e s t i o n o f th e p r o b a b i l i t y ( n o t " m e a s u r e " ) o f V , a n d i n d e e d , t h e m o r e d e l i c a t e
q u e s t i o n o f i ts s u b s e t B 0 , w a s a c c e s s i b le t o h is n e w t h e o r y o f " d e n u m e r a b l e
p r o b a b i l i t y " . W h e n o n e c o n s i d e r s t h a t V A N V L EC K 'S m a i n g o a l w a s t o c o n s t r u c t a
n o n m e a s u r a b l e s e t ( th e v e r y e x i s t e n c e o f w h i c h w a s d i s c o m f o r t i n g to BOREL), h e
o p p o r t u n i t y o f f e r e d B O R EL t o g e t a b e t t e r r e s u l t b y h i s " m o r e e f f e c ti v e " t h e o r y
m i g h t w e l l h a v e d i r e c t e d h is a t t e n t i o n a l o n g t h e t r a i l le a d i n g t o h is f o r m u l a t i o n o f
t h e S t r o n g L a w .
6. Borel's Chapter III: Continued Fractions
6 . 1 . T h e S e t t i n g o f t h e P r o b l e m
I n C h a p t e r I I I, B O R EL r e t u r n s t o c o n t i n u e d f r a c ti o n s , th e s u b j e c t o f h i s e a r li e r
4 5 - p a g e p a p e r o f 19 03 , " C o n t r i b u t i o n fi l ' A n a l y s e A r i t h m 6 t i q u e d u C o n t i n u . "
T h e c o n t i n u e d f r a c t i o n e x p a n s i o n o f an i r r a t io n a l n u m b e r x i n [ 0, 1 ],
1x - -
a l + l
a 2 + l
a 3 + l
w h e r e e a c h e l e m e n t a , = a , ( x ) is a p o s i t iv e i n t e g e r , is in s o m e w a y s p a r a l l e l t o t h a t o f
t h e d e c i m a l ( o r q - a ry ) e x p a n s i o n . B O R E L c o n s i d e r e d i t as a n i n s t a n c e o f a s e q u e n c e
o f in f i n it e ly m a n y t ri a ls ( o n e f o r e a c h i n t e g e r a n) e a c h o f w h i c h m a y h a v e i n f in i t el y
m a n y o u t c o m e s , e .g . G m a y e q u a l 1, 2 , 3 , . . . , k . . . .
T h e p r o b a b i l i t i e s P~,k, th a t a~ = k , sa t i s fy
~ pi ,k = l .k = l
O n e c o u l d , a pr ior i , m a k e a r b i t r a r y h y p o t h e s e s i n a d d i t i o n , b u t B O R E L a s s e r t e d ,
. .. ; w e a r e g o i n g t o s t u d y t h e h y p o t h e s e s t o w h i c h o n e i s l e d w h e n t a k i n g t h e
g e o m e t r i c p o i n t o f v i e w a l re a d y i n d i c a t e d /t p r o p o s t h e d e c i m a l n u m b e r s .
BOREL (1909: 264) .
T h i s m e a n s , i n e f f e ct , t h a t t h e p r o b a b i l i t y f o r x t o l ie in a n y s u b - i n t e r v a l o f [ 0, 1 ]
is th e l e n g t h o f t h a t s u b - i n t er v a l. 1 O n c e a g a i n t h e m o t i v a t i o n f o r th e e x a m p l e c o m e s
f r o m p r o b l e m s o f " g e o m e t r i c " o r " c o n t i n u o u s " p r o b a b i l i t y , in t h e m o s t n a i v e se ns e.
1 B y e x t e n s i o n , i t a l s o m e a n s t h a t t h e p r o b a b i l i t y t h a t x l i e s i n a B O R E L s e t o f [ 0 , 1 ] i s t h e
m e a s u r e o f t h a t s e t , b u t t h i s a ll i m p o r t a n t a n d n o n - n a i v e e x t e n s i o n o f g e o m e t r i c p r o b a b i l it y b y
m e a n s o f m e a s u r e t h e o r y i s n o t e m p l o y e d o r e x p li c it ly a c k n o w l e d g e d i n t h e p a p e r o f 19 09 . S e e
F A BE R 'S r e m a r k (§ 9 .2 ), m a d e a f t e r r e a d i n g t h e p a p e r o f 1 9 09 , w h i c h e x p l i c i t ly q u e s t i o n s w h e t h e r s u c h
a n e x t e n s i o n c a n b e m a d e o f " d e n u m e r a b l e p r o b a bi l it y " .
a l t e r n a t i v e o f c o n s i d e r a b l e w e i g h t m u s t b e c o n s i d e r e d . T h e r e l e v a n t fa c t is t h a t t h e
S w e d i s h m a t h e m a t i c i a n A . W I M A N h a d o b t a i n e d a n a n a l o g o u s r e su l t d a t in g b a c k
t o 1 9 0 0 - 1 9 0 1 w h i c h a s s e r t s z e r o p r o b a b i l i t y f o r a s e t i n th e u n i t i n te r v a l . W IM A N 'S
s e t ( a s u b s e t o f t h e i r r a t i o n a l s i n t h e u n i t i n t e r v a l ) w a s a l s o d e f i n e d b y as o p h i s ti c a te d c o n d i t i o n o n t h e a s y m p t o t i c b e h a v i o r o f t h e t e rm s a n ( x ) o f t h e
e x p a n s i o n o f i ts m e m b e r s i n c o n t i n u e d f r a c t io n s . T h e c a l c u l a ti o n s s h o w i n g i t t o
h a v e z e r o " p r o b a b i l i t y " o v e r l a p p e d s e v e r a l o f B O R EL 'S . W I M A N ' s p r o o f i n v o l v e d
a n e x p l ic i t u se o f B O R EL 's n e w t h e o r y o f m e a s u r e t o e x t e n d t h e s c o p e o f c l as s ic a l
g e o m e t r i c p r o b a b i l i t y a n d r e li e d o n a s c r u p u l o u s u s e o f c o u n t a b l e s u b - a d d i t i v it y o f
s u c h m e a s u r e i n t h e p r o o f (WIMAN ( 1900 ; 1901)). WIMAN h i m s e l f was r e pa i r i ng
s e r i o u s d e f e c t s in t h e p r e v i o u s w o r k o f h i s c o l l e a g u e T . B R O D t~ N (1 90 0), i n a p o l e m i c
t h a t r a g e d b e t w e e n t h e t w o . T h e p r o b l e m t h e y d e b a t e d h a d a r is e n in t h e c o n t e x t o f
C e l e s t i a l M e c h a n i c s w h e r e i t h a d b e e n r a i s e d b y t h e S w e d i s h m a t h e m a t i c i a n -
a s t r o n o m e r H . G Y L DI2 N; a s a re s u l t t h e i r p o l e m i c w a s d i s p u t e d i n j o u r n a l s r e a d b y
N o r d i c m a t h e m a t i c a l a s t r o m e r s b u t w a s h a r d l y f am i li ar t o t h e g e n e r a l E u r o -
p e a n m a t h e m a t i c a l c o m m u n i t y . I n c o n t r a s t t o t h e c o r r e s p o n d i n g c a s e o f V AN
V L EC K 'S w o r k o n d y a d i c d ig it s, t h e r e is n o d o u b t o f BO R EL 'S a c q u a i n t a n c e w i t h
t h i s p r i o r w o r k ; i n 1 9 0 5 BO R E L e x p l i c i t y g a v e W I M A N p r i o r i t y f o r t h e i n -
t r o d u c t i o n o f m e a s u r e i n to g e o m e t r i c p r o b a b i li ty . T h i s a c k n o w l e d g e m e n t , a c c o m -
p a n i e d b y a b i b l i o g r a p h i c r e f e r e n c e t o W I M A N ' s p a p e r s , o c c u r s i n B O R EL (1 90 5),
w h i c h i s d i s c u s s e d b e l o w ( c f § 7 .3 ). Ho w ev e r , BOREL do es no t , i n 1905 , r e f e r t o
t h e c o n t e n t o f W I M A N ' s re s u lt , n o r d o e s h e e v e n m e n t i o n t h e o c c u r r e n c e o f
c o n t i n u e d f r a c t io n s i n t h e f o r m u l a t i o n o f t h e p r o b l e m s o l v e d b y W IM A N . T h e
m u c h m o r e i n f lu e n t i a l p a p e r o f 1 90 9 la c k s a n y r e f e r e n c e to W I M AN 'S w o r k
w h a t s o e v e r , c o n s i s t e n t w i t h it s o m i s s i o n o f a n y i n t e r p r e t a t i o n i n t e r m s o f
m e a s u r e o f t h e r e s u l t a b o u t c o n t i n u e d f r a c t io n s .
B O RE L s aw h i s C o n t i n u e d F r a c t i o n T h e o r e m a s a n e x a m p l e o f a g e n e r al t h e o r y
o f p r o b a b i l i t y o n a b s t r a ct s ets (t h e p r e s u m e d m e a n i n g o f " p o i n t e d e v u e l o g i q ue " ) .
O n t h e e v i d e n c e p r e s e n t e d a b o v e , h e w a s a t b e s t u n c l e a r t h a t h is g e n e r a l t h e o r y w a s ,
i n th i s c a s e, e q u i v a l e n t t o m e a s u r e t h e o r y . I t i s e n t i r e l y p o s s i b le h e t h o u g h t
d e n u m e r a b l e p r o b a b i l i t y a p p l i e d t o c o n t i n u e d f r a c t io n s o f f er e d a d is t in c t , m o r e
" e f f e c t i v e " (i.e., " c o n s t r u c t i v i s t " ) a l t e r n a t i v e . I n a n y c a s e B O R E L u n q u e s t i o n a b l y
k n e w o f a t l e a st o n e p r o b a b i l i s t i c r e s u lt c o n c e r n i n g c o n t i n u e d f r a c ti o n s b e f o r e 1 9 09
a n d t h is c o u l d w e l l h a v e b e e n t h e g e n e s i s o f h is o w n e x a m p l e o f t h e g e n e r a l t h e o r y .
T h e d i s t i n c t i o n b e t w e e n W I M A N a n d V A N VLECK a s p r e c u r s o r s o f B O RE L ism a r k e d i n tw o w a y s : f ir st , w h i le b o t h e m p l o y e d m e a s u r e t h e o r y , WIMAN,u n l i k e
VAN VLECK, e x p l i c it ly c o n s i d e r e d h is t h e o r e m a s a s o l u t i o n t o a p r o b l e m o f
p r o b a b i l i t y ; s e c o n d , w h i le V A N V L E C K 's w o r k m a y h a v e b e e n k n o w n t o BOREL,
WIMAN's u n q u e s t i o n a b l y w a s k n o w n .
6 .5 . T h e C a n t e ll i M o d i f i c a t io n o f t h e B o r e l Z e r o - O n e L a w
T h e B O R EL Z e r o - O n e L a w h a s u n d e r g o n e c o n s i d e ra b l e g e n e r a l i z a ti o n s in c e its
f ir st f o r m u l a t i o n i n 19 09 . I n p a r ti c u l a r , th e " z e r o " c a s e ( A ~ = 0 w h e n ~ p ,
c o n v e r g e s ) h a s s u b s e q u e n t l y b e e n e s t a b l is h e d w i t h n o a p p e a l t o t h e i n d e p e n d e n c e
o f t h e e v e n t s w h o s e p r o b a b i l i t i e s a r e p 1, P 2 , . .- - T h e s e e v e n t s a r e t h e " t r i a l s " o f t h e
o r i g i n a l BO R EL f o r m u l a t i o n . T h i s r e m o v a l o f t h e h y p o t h e s i s o f i n d e p e n d e n c e i n t h e
I f w e r e - i n t roduc e t he no t a t i o n P(En)=p, an d A~o=P(l imsupE,) , t he n w ecO
ob tain the CANTELLI mo dif icat i on of BOREL's resul t : ~ Pn conv erges imp l ies A~o
= 0 , w i t hou t a ny a s s um pt i on a s to i nde pe nde nc e .In § 10.2 below, as wel l as in § 7 imm edi ate l y fol low ing we give ins tances of this
reas oni ng by BOREL, FRt~CHET, and HAUSDORFF, a l l of wh om pre ced ed
C A N T E L L I .
7. Borel ' s Earl ier Works
7.1. Introduction
This sec t ion dea l s wi th two ea r l i e r works of BOREL: "Contr ibut ion & l 'analyse
a r i thm 6t ique du co nt in u" (BOREL (1903) ) and " Re m arq ue s sur ce r t a ines ques t ions
de probabili t6" (BOREL (1905)).
Each of these works wi ll be re la ted to the cons ide ra t ions of BOREL'S "L es
p roba b i l i t 6 s d6nombra b l e s e t l e u r s a pp l i c a t i ons a r i t hm6t i que s " i n o rde r t o g i ve
• fur the r ev idence su ppo r t ing the ana lys is of BOREL (1909) presen ted above .
In the discuss ion of BOREL (1903), the focus wi l l be on the inf ini te sub-additivity
of geom et r i ca l vo lum e. In pa r t i cu la r , we sha ll focus on the resu l t tha t , for su i tab le
sets E, El , E2 . . . . in Eu cl id ean n-d ime nsio nal space, the assum ption s ~ vol(E~) < 0%1
and E c l i ra sup Ei , imp ly v ol (E )= 0. (This i s the resu l t of 1909, nam ely, A~ = 0 in
the case of converg ence , except in geo met r i c an d n ot p robabi l i s t i c t e rms .) The
a bs e nc e o f a ny a s s umpt i ons c o r r e s pon d i ng t o i nde pe nd e nc e i s no t e w or t hy , i.e. this
beco mes a "CANTELLI" resu l t when p ut in a probabi l i s t i c se t ting . I t is prove d b y
BOREL, us ing "CANTELLI- l ike" reason ing a nd some geom et r i c hypoth eses on the
se ts E , E 1, E2 , .. . suf fi c ien t to ensure tha t they h ave "v ol um e" in an e lem enta ry
sense.
I t wi ll be show n, by c i ta t io ns fro m BOREL (1903), that cou nta ble su b-add i t ivi ty,
the ke y to BOREL's pr oo f of the abov e resu l t, was in t imate ly conn ec ted in BOREL's
mind wi th the HEINE-BOREL Th eore m. On th i s ev idence it s eems a t l eas t h ighly
plaus ib le tha t BOREL's re luc tance to employ countable addi t iv i ty and sub-a dd i t i v it y i n de num e ra b l e p roba b i l i t y w a s be c a us e o f t he s t r a nge ne w c on t e x t i n
which the HEINE-BOREL Th eo rem was inappl i cable . Because cou ntab le add i t iv i ty
a nd s ub -a dd i t iv i t y ha d no t p l a ye d a r o l e i n e i the r f in i te o r c on t i nuous p roba b i l i t y
by 1909 1, there was no s t rongly sugges t ive evidence that e i ther was an essent ia l
p r o p e r t y t o d e m a n d o f " d e n u m e r a b l e p r o b a b i l it y " . I n d e pe n d e n c e, o n t h e o t h e r
hand, was a lmos t the ch arac te r i s t i c fea ture of probab i l i ty , and the ex tens ion of
i nde pe nde n c e t o t he de num e ra b l e c a s e e xe r te d a c o r r e s pond i ng l y s tr onge r a ppe a l
to BOREL as the essent ia l ingredient for his new theory in 1909.
In s umm ary , the conten t s of BOREL (1903) he lp con s ide rably in acc oun t ing for
BOREL's t imidi ty about countable sub-addi t iv i ty ( in 1909) , and in provid ingreasons for h i s needles s ly res t r i c t ed as se r t ion about A~ in the convergent case .
* The e xcept ion a l d iscu ss ion of con t in uou s probab i l i ty by WIMAN (1900; 1901) re fe rred to abov e
T h e p a p e r B O R E L ( 19 05 ) r e p r e s e n t s B O R EL 's c h i e f c o n t r i b u t i o n t o p r o b a b i l i t y
p r i o r t o 1 90 9. T h i s p a p e r r e p r e s e n t e d ( fo r al l i ts s h o r t c o m i n g s ) a s i g n i f i c a n t a d v a n c e
i n t h e f o u n d a t i o n s o f p r o b a b i l i t y t h e o r y w h i c h h a s b e e n u n a c c o u n t a b l y n e g l e c te d .
I t p r o p o s e d t h a t i n t h e u n i t i n t e rv a l t h e m e a n i n g o f (g e o m e tr ic ) p r o b a b i l it y ,i d e n t i f i e d w i t h l e n g t h , b e s u b s t a n t i a l l y g e n e r a l i z e d t o b e i d e n t i f i e d w i t h h i s n e w
t h e o r y o f m e a s u re .
B O R E L (1 90 5) u n d e r s c o r e s h o w s i g n i f i c a n t w a s t h e g a p b e t w e e n B O R E L's v ie w o f
( g e o m e t r i c ) p r o b a b i l i t y o n t h e u n i t i n t e r v a l , o n t h e o n e h a n d , a n d h i s v i e w o f
d e n u m e r a b l e p r o b a b i l i t y o n t h e o t h e r . W i t h B O R E L (1 90 5) i n m i n d o n e c a n b e
a l m o s t c e r t a i n t h a t B O R E L 'S r e s e r v a t i o n s ( i n 1 9 0 9 ) a b o u t t h e r o l e o f c o u n t a b l e
a d d i t i v i ty i n " d e n u m e r a b l e p r o b a b i l i t y " s t em f r o m a n i n c o m p l e t e r e c o g n i ti o n t h a t
t h e m a c h i n e r y o f t h e " g e o m e t r i c p o i n t o f v i e w " , a p p li c a b le i n h i s d e c i m a l a n d
c o n t i n u e d f r a c t i o n e x a m p l e s , s h o u l d b e a v a i l a b l e a s w e l l i n t h e g e n e r a l t h e o r y .
T h e a i m o f t h e d e t a i l e d d i s c u s s i o n o f t h e s e e a r l ie r w o r k s i s t h u s t o s h e d l i g h t o n
t h e s h o r t c o m i n g s o f B O R E L (1 90 9) ( n o t e d a b o v e ) b y c o m p a r i n g t h e m w i t h
v i e w p o i n t s B O R E L h i m s e l f p o s s e s s e d b e f o r e 1 90 9.
7 .2 . Bore l (19 03 ) : F oreshadowings o f Can te l l i-L i ke Reason ing
I n B O R E L (1 90 3), c o n t i n u e d f r a c t i o n s h a d b e e n t h e m a i n o b j e c t o f s t u d y a n d n o t
m e r e l y a s o u r c e , a m o n g o t h e r s , o f e x a m p l e s a s t h e y w e r e i n B O R E L ( 19 09 ). T h e
g e n e r a l p u r p o s e o f B O RE L (1 90 3) w a s t o e s t a b l i s h t h e e x i s te n c e o f c o v e r i n g s o f t h e
i n t e r v a l [ 0 , 1 ] b y s u b - in t e r v a l s , m e m b e r s h i p i n w h i c h d e m a n d s a h i g h d e g r e e o f
a p p r o x i m a b i l i t y b y r a t i o n a l s . A t y p i c a l s u c h r e s u l t is , e.g., t h a t every r e a l n u m b e r c~
p o s s e s s e s r a t i o n a l a p p r o x i m a n t s P /Q s u c h t h a t
- ~ < 1 /~ Q 2
wh ere (2 can be req u i red to l ie in a p r e sc r ibe d in t e rv a l (A , B ) , sa t i s fy ing
I O < A < 1 5 A 2 < B .
I n c o n s e q u e n c e , o n e c a n p r e s c r i b e a n i n f i n it e s e q u e n c e o f in t e r v a l s ( A , , B n),s a t i s f y i n g t h e a b o v e , a n d t h e r e w i l l e x i st a s e q u e n c e o f c o r r e s p o n d i n g a p p r o x i m a n t s
T h e m a i n t o o l u s e d i n B O R EL ( 19 03 ) is t h e r e p e a t e d u s e o f f i n it e o r d e n u m a r a b l e
c o v e r s o f [-0 , 1 ] ( o r i ts n - d i m e n s i o n a l a n a l o g , t h e n - d i m e n s i o n a l c u b e) . T h e s e c o v e r s
a r e u s e d t o a s s e r t r e l a t i o n s b e t w e e n t h e l e n g t h s o f t h e c o v e r i n g i n t e r v a l s a n d t h e
l e n g t h ( o r , i n h i g h e r d i m e n s i o n s , v o l u m e ) o f s e ts w h i c h a r e e i t h e r c o n t a i n e d i n a
f in i t e subco ve r , o r , a l t e rn a t ive ly , cove re d in f in i t e ly o f t en .
T h e s e n o t i o n s h a d a l r e a d y b e e n e x p l o it e d b y B O R E L i n b o t h h i s c e l e b r a te d
thesis (BOREL (1895)) an d in h is Lemons sur la Th~orie des Fonct ions (BOREL (1898)).
W h a t i s n o w c a ll e d th e H E IN E -B O RE L T h e o r e m h a d a l r e a d y b e e n s t a t e d b y B O R E L,
f o r d i m e n s i o n n = 1, i n h i s t h es i s a n d i n B O R E L ( 18 98 ). T h e r e is n o q u e s t i o n t h a t t h e
H E I N E -B O R E L T h e o r e m ( in o n e d i m e n s i o n ) w a s c e n t r a l t o B O R EL 's d e f i n i ti o n o f
m e a s u r e , a n d t o t h e d e f i n i ti o n o f t h o s e s e ts ( s in c e c a ll e d " B O R E L - m e a s u r a b l e " ) t o
w h i c h h e a p p l i e d t h i s t h e o r y o f m e a s u r e . A l t h o u g h B O R EL f a il e d t o g e n e r a l i z e hi s
t h e o r y o f m e a s u r e t o h i g h e r d i m e n s i o n s , h e d i d g e n e r a l i ze t h e H E I N E -B O R E LT h e o r e m , a n d f r o m i t h e d e d u c e d a f o r m o f c o u n t a b l e s u b - a d d i ti v i ty i n h ig h e r
d i m e n s i o n s . T h e e x t e n s i o n o f t h e H E I N E -B O R E L T h e o r e m ( u si n g o n l y o p e n c o v e r s
o f a n e s p e c i a l l y s i m p l e s o r t ) o c c u r r e d i n B O R E L (1 90 3). T h e r e h e g a v e t h e e x t e n s i o n
( T h e o r e m V I I I ) t o b o u n d e d c l o s e d s et s i n d i m e n s i o n n , r e s tr i c ti n g h i m s e l f t o
d e n u m e r a b l e c o v e r s. (B O R EL l a t e r a c k n o w l e d g e d L E BE SG U E 's g e n e r a l i z a t i o n o f t h e
r e su l t to a p p l y t o n o n - d e n u m e r a b l e c o v e r s as w e ll .)
O f s p ec i a l i n t e r e s t is t h e o c c u r r e n c e o f t h e t h e o r e m o n v o l u m e s s t a t e d a b o v e .
T h i s i s T h e o r e m X I b is i n t h e n o t a t i o n o f B O R E L (1 90 3). A s r e m a r k e d i n th e
i n t r o d u c t i o n , i t is a s p e c ia l c a s e o f w h a t is n o w c o n s i d e r e d t h e " C A N T E L L I " p a r t o f
the BOREL-CANTELLIL e m m a s .
T h e s p e c i a li ti e s t h a t s u r r o u n d t h is t h e o r e m o f BO R EL l ie in t h e a s s u m p t i o n s ,
d e r i v i n g f r o m t h e c o n t e x t o f t h e p a p e r o f 19 03 a s a w h o l e , t h a t E l , E 2 . . . . . En, . . . a r e
" d o m a i n s " i n s o m e f ix e d E u c l i d e a n s p a ce s, R k, o f d i m e n s i o n k . A " d o m a i n " is
d e f i n e d as a c l o s e d b o u n d e d p a r t o f k - s p a c e d e t e r m i n e d b y a f in i te n u m b e r o f
a l g e b r a i c i n e q u a l i t i e s
qSj(x 1, . - . , xk) > 0 j = 1, 2 . . . . M .
T h e v o l u m e o f a d o m a i n is a n n - d i m e n s i o n a l ( RIE M A N N) i n t e g ra l , w i th
i n t e g r a n d 1, o v e r th e i n t e r i o r o f t h is d o m a i n . B O R EL c o n s i d e r e d o n l y d o m a i n sw h i c h , i n m o d e r n t e r m i n o l o g y , h a v e n o n - e m p t y i n t e r i o r s a n d w h i c h a r e t h e
c l o s u r e s o f t h e i r i n t er i o r s. I n p a r t i c u l a r , e v e r y d o m a i n c o n s i d e r e d h a s a s t ri c tl y
p o s i t i v e v o l u m e . ( T h u s , f o r e x a m p l e , c l o s e d s p h e r e s a n d c l o s e d c u b e s a r eco
" d o m a i n s " . ) B O R E L s h o w e d t h a t i f ~ v o l ( E , ) is c o n v e r g e n t , t h e n l im s u p E ~ = H1 8 4 o o
h a s " s m a l l c o v e r i n g " . S p e c if ic a ll y , g i v e n e > 0 , t h e r e a r e d o m a i n s
H 1, H a . . . . . H . . . . . s u c h t h a t
H c U ( in t er io r H n), a n d ~ v o l ( H , ) < e .
1 1
T h i s m a y b e c a l l e d B O R E L 'S v e r s i o n o f t h e C A N T E L L I L e m m a . B O R E L's o r i g i n a l
f o r m u l a t i o n w i l l b e p r e s e n t e d b e l o w .
W e n o t e t h a t B O RE L s t o p p e d s h o r t o f c o n c l u d i n g v o l ( H ) = 0 . I n d e e d h e c o u l d
n o t d o t h i s si n ce th e p o i n t s e t H n e e d n o t b e a d o m a i n , a n d s o v o l ( H ) n e e d n o t e x i s t
a s a R IE M A N N i n t e g ra l . A t t h e d a t e o f t h is p a p e r t h e t h e o r y o f m e a s u r e h a d n o t b e e n
g e n e r a l i z e d t o n - d i m e n s i o n a l E u c l i d e a n s p a c e, n o r h a d t h e th e o r y o f t h e
L E B ES G U E i n t e g r a l b e e n e s t a b l i sh e d .
A p a r t f r o m t h e re s t ri c ti o n t o " d o m a i n s " t h e t h e o r e m is t h u s v e r y m u c h t h e
C A N TE LL I L e m m a . W h a t is e m i n e n t l y r e m a r k a b l e is t h e l in e o f r e a s o n i n g
e m p l o y e d b y B O R EL . T h e " C A N T E L L I " p r o o f g i v e n in § 6.5 a b o v e , w h e n s p e c i a li z e dt o t h e c a s e a t h a n d , p r o v i d e s t h e d e s i re d s e t o f d o m a i n s H 1 H 2 , . . . , H . . . . . a s
H i = E N + i w h e r e N i s c h o s e n so t h a t ~ v o l ( E N + i ) < e .i = 1
T h e i n t e re s t i n g q u e s t i o n i s h o w B O R E L a r r a n g e d t o p r o v e t h i s t h e o r e m . T o
e x a m i n e t h i s , i t i s n e c e s s a r y t o r e a d h i s e x a c t f o r m u l a t i o n o f t h i s r e s u l t a n d i ts
i m m e d i a t e p r e c u r s o r , T h e o r e m X I .
T h e o r e m X I . L e t E b e a d o m a i n a n d E l , E 2 , . . . , E h . . . . d o m a i n s s u c h t h a t e v e r y
p o i n t o f E i s i n t e r io r t o a n i n f i n i t y o f t h e m ; t h e n o n e c a n a s s e r t th a t t h e v o l u m e s
v ~ , v 2 , . . . , V h, . . . o f t h e s e d o m a i n s a r e s u c h t h a t t h e s e r i e s
V JC- V2 ~ - . . . -Jr Vh ~- . . •
i s d i v e r g e n t .
T h e o r e m X I bis . L e t E l , E 2 , . . . , E h . . . . . d o m a i n s w i t h v o l u m e s V l , v 2 , . . . , V h, . ..
b e s u c h t h a t t h e s e r i e s
V -}- V 2 -Jf- . . . ~- Uh @ . . .
i s c o n v e r g e n t ; t h e n o n e c a n a s s e r t t h a t t h e s e t H o f p o i n t s w h i c h b e l o n g t o a n
i n f i n i t y o f t h e s e d o m a i n s i s s u c h t h a t , b e i n g g i v e n e a r b i t r a r i l y s m a l l , o n e c a n
c o n s t r u c t d o m a i n s H a , H 2 . . . . . H . . . . . . f i n i t e o r d e n u m e r a b l y i n f i n i t e in n u m b e r ,
s u c h t h a t e v e r y p o i n t o f H i s i n t e r io r t o o n e o f t h e m a n d t h a t , in a d d i t i o n , V~ b e i n g
t h e vo l u m e o f H ~ o n e h a s
v l + v 2 + . . . + v ~ + .- . < e .
( I t a l i c s i n t h e o r i g i n a l . ) B O R E L ( 1 9 0 3 : 3 6 2 ) .
T h e f u l l l in e o f d e v e l o p m e n t l e a d i n g t o th e s e t h e o r e m s p r o v i d e u n a s s a i l a b l e
e v i d e n c e a s to h o w B O R E L, i n t h i s c o n t e x t , a s s o c i a t e d s u b - a d d i t i v i t y a n d t h e
" C A N T E L L I L e m m a " w i t h t h e H E I N E - B O R E L r e su l t . T h i s l in e o f d e v e l o p m e n t i s t h e
f o l l o w i n g c h a i n o f t h e o r e m s , o c c u r r i n g i n S e c t i o n 1 9 o f B O R E L (1 9 03 ).
T h e o r e m V I I I . L e t E b e a g i v e n c l o s e d b o u n d e d s e t, a n d E l , E 2 , . . . , E p . . . . a
d e n u m e r a b l e i n f i n i t y [ f o o t n o t e o m i t t e d ] o f s e t s s u c h t h a t e v e r y p o i n t o f E i s
I N T E R I O R t o a t l e a s t o n e o f t h e m ; i t i s p o s s i b l e t o f i n d a m o n g E l , E 2 , . . . , E p , . . .
a F I N I T E n u m b e r o f s e t s s u c h t h a t e v e r y p o i n t o f E is i n t e r i o r t o a t l e a s t o n e o f
t h e m . ( I t a l ic s i n t h e o r i g i n a l . ) B O R E L ( 1 9 0 3 : 3 5 7 ) 1
This is as c lear a rende ring o f the n-d imen sional HEINE-BOREL heo rem as could b e wished. BOREL
even supplemented i t wi th an extens ion to se t s which a re only c losed (or bounded) a f te r a su i table
projec t ive t ransformat ion . ( In mode rn terms, he cons ide red projective 2-space, or more generally, n-
space.) Such sets he cal led projectively closed o r projectively bounded.
He then g ave a s imple example of three se ts in the pro jec t ive p lane w hich a re projectively closed and
projectively bounded and col lec t ive ly cover the p lane ( two d imensions b e ing chosen for ease of
expos it ion). Thus he could ap ply The orem VII I to each of these se ts . BOREL hen s ta ted a "gen era l ized"HEINE-BORELt he o re m:
The ore m V I I I b i s. If one has a denumerable inf ini ty of sets El , E 2 , . . . , E n , . . . such that every
point of the plane is in the I N T E R I O R of at least one of them (the points at infinity included, ofcourse), one can determine among the E i a fin ite number o f sets such that every point of the plane
is interior to one of them. (Italics in th e original.) BOREL (1903: 359).
Thus BOREL was aw are tha t the se t t ing of the HEINE-BOREL heorem could b e widened f rom n-
dimensions to a t l eas t ce r ta in a l te rna t ive spaces w i thout ch anging the na ture of the asser tion .
N e x t , r e s t r i c t i ng c ons i de r a t i on t o doma i ns a nd t he i r a s s oc i a t e d vo l ume s ,
BOREL es tabl ished his f i rs t (and key) resul t for sub-addi t ivi ty:
T h e o r e m I X. Wh e n a d o ma i n E i s s u c h t h a t e a c h p o i n t i s i n t e r i o r t o a d o ma i n E i( i = 1 , 2 , 3 . . . . , n . . . ) , one can asser t tha t the s um o f the v o lum es o f the d oma ins E i i s
g r e a t e r t h a n t h e v o l u m e o f E . (Ital ics in the original.) BOREL (1903: 360).
I n o t he r w ords ,
impl ies
oo
E c U ( inter io r(E0)1
oo
vol(E) < y~ vol(E3.1
Th e pro of , to be supp l i ed by the reader , involves the use of the HEINE-BOREL
t h e o r e m ~Th eo re m VII I ) a s a pre l imin ary to as sure the ex i s tence of an N such tha t
N
E c ~) ( inter io r (El)).1
F rom t h i s one c onc l ude s (p r e s uma b l y by e l e me n t a ry c a l c u l us )
N
vol(E ) < ~ vol(Ei)1
and the resul t fol lows.
T he s t a t e me n t o f T h e or e m IX is to be i n t e rp r e t e d a s i nc lud i ng t he c a s e
oo
vol(E /) = + oo.1
BOREL then poin ted out tha t one can g ive th i s same asse r t ion an occas iona l ly
more c onve n i e n t f o rm a s fo l l ow s :
T he ore m IX b i s. Gi v e n a d o ma i n E a n d a d e n u me r a b l e i n f i n i t y o f d o ma i n s
E l , E 2 , . . . , E n , . ..
s u c h t h a t o n e h a s
• vol(Ei) <vol(E),1
t h e n t h e r e a r e p o i n t s o f E n o t i n t h e i n t e r i o r o f a n y E i. (Ital ics in the original.)BOREL (1903: 361).
Thi s i s, o f course , no m ore than a cont ra pos i t ive fo rmu la t ion o f the or ig ina l
T he ore m IX , r e qu i r i ng no fu r t he r p roo f .
T h i s pa r t i c u l a r f o rmul a t i on c a n be i mme d i a t e l y s ha rpe ne d by us e o f a n
e lem enta ry a r gum ent w hich em ploys s l igh tly l a rge r doma ins E~ conta in ing El .
T he s ha rpe ne d fo rmul a t i on i s :
T h e o r e m X . I f E has vo lume v, and a denumerab le infinity of domain s E i
( i = 1, 2, . . . , n, . . . ) having volumes vl, are such t h a t
~ v i < v1
the there are points o f E b elonging to none of the E i . (Ital ics in the original.)
BOREL (1 90 3:3 61 362).
Theorem X has a s l igh t ly s t ronger conc lus ion than i t s immedia te predecessor
s ince i t avoids re fe rence to the " in te r ior s " of the se ts E~. Thi s n on- to polo gica lve rs ion i s the one which s t r ikes the contempora ry reader as foreshadowing the
genera l i za t ion to measure theory (or probabi l i ty ) . As we have shown, however , i t
w a s a c h i e ve d on l y a f te r T h e or e m IX bis ( involv ing " in te r iors " ) , and th i s in turn
depended c ruc ia l ly on a HEINE-BOREL argument involv ing "open covers" .
T h e o r e m X I a n d X I bis, c i t ed above , a re now d i rec t consequences of the
preceding. BOREL leaves thei r proofs to the reader .
Summ ar iz ing , BOREL wel l knew tha t in ce r t a in c i rcum s tances i f the se t s E ko0
satisfy ~ m(Ek) < oe, th en th e set H = l im sup Ek, of those p oin t s in inf in i te ly m any of
1the Ek'S, has covers of a rb i t ra r i ly smal l to ta l vo lum e. He d id not s t a t e th i s in the
genera l i ty of measu re theory . Th ere i s no reason to suppo se tha t BOREL so muc h as
c onc e i ve d o f " a bs t r a c t i n g" me a s u re t he o ry t o t he de g re e o f ge nera l it y ne e de d fo r
the CANTELLI imp rov em ent of h i s Zero- On e Law. Indeed , he d id n ot even cons ide r
the s t ra ight forw ard genera l i za t ion of measu re theo ry f rom 1 to n d imens ions . Even
i f he had conce ived of n-d imens ion a l measure , the remain in g s t ep to an ab s t rac t
me asu re re ma ins imm ense. In pa r t icular , the absen ce of a HEINE-BOREL resu l t in
the abs t rac t case might have proved an unbr idgeable gul f , for i t was th i s resu l t
which was bas ic to BOREL's meth ods of proof . The conc lus ion seems inescapable
tha t BOREL knew the "CANTELLI" theo rem in a geom et r i c se t ting only , where
topolo gica l cons ide ra t ions were re levant and ava i l ab le . The l ack of the nota t ion for
l im sup E k for a co l l ec t ion of event s fur the r d i sgui sed the resemblance be tween
the se t H of h i s paper of 1903 and the no wh ere des igna ted se t ( ° 'i n fin it ely ma ny
successes" ) in the paper of 1909 whose probabi l i ty i s A~ .
We a re thus l ed to add an ad di t iona l sp ecula t ion co ncern in g BOREL'S t rea tm ent
of h i s Zero -On e Law : the ab sence of a topolog y in the space of tr i a ls prev ented h im
from es tab l i sh ing a HEINE-BOREL Th eo rem ; th is absence in turn b lock ed the way
for h i s use of coun table sub-ad di t iv i ty in the se t ting of " probab i l i t6 d6n om brab le" ,
e ve n i f he ha d t hough t o f p roba b i l i t y a s a na l ogous t o vo l ume (o r me a s u re ) .
In v iew of the Theorem XI h i s c i t ed above , in which sub -addi t iv i ty i s proved , we
are s im ul tan eou sly led to the surpris ing con clus ion tha t BOREL was as c lose in 1903
to a r igo rous p roo f of the "CANTELLI" ve rs ion of h i s Zero- On e L aw as was
CANTELLI in 1917. The difference is that CANTELLI's work deals explicit ly with
prob abi l i ty a nd so was t aken up by succeeding probabi l i s t s, whi le BOREL'S pap er of
Inc identa l ly , BOREL regards the de te rmina t ion P(E)=0 for the ra t iona l s a s
c l e a rl y " c o r r e c t " w i th no s us t a in i ng a rgum e n t o t he r t ha n a ppe a l t o t he a u t ho r i t y o f
POINCARI~, wh o had inde ed rega rde d i t as evident , before the existence of a t he o ry
of measure . W e h ave ex am ine d POINCARI~'s lec ture n otes , Calcul des Probabilitds(1893-1894 lectures), a t the Ins t i tu t Henr i Poincar6 , Pa r i s , where th i s en igmat ic
p ron oun c e m e n t i s t o be found , a l r e a dy i nc o rpo ra t e d i n the t e x t a t tha t " p r e m a t u re "
da te , some 16 years be fore the publ i ca t ion of h i s Calcul des Probabilit&
3) An exam ple is considered, n am ely th e se t E ("), def ined for each inte ger valu e
of the p a ram ete r n as a ll rea l numb ers e in [-0 , 1] such tha t the re a re re la t ive ly pr ime
integers p, q satisfying
<1q 1 q"
For each in teger va lue of n , i t i s ev ident tha t
E (") c ~ l~)qw he re
I~½ = p 1 p t- ,q" ' q
p, q are re la t ively pr ime, i.e., (p, q) = 1, an d the u ni on is ov er all such pairs p and q. It
i s then as se r t ed (wi thout comment ) tha t
o~ 2
)=q 2 -P(E(')) < 2 1(I~½ O(q) q.(p , q ) = 1
He re ~b(q) = the n um be r of integers less tha n and re la t ively pr im e to q, and the ser iesclear ly con verg es i f n > 2. This i s a wond erfu l ly c lear ex amp le o f BOREL's expl ic i t
use of coun table sub-addi t iv i ty , bu t only in the contex t of geom et r i c probabi l i ty , o r
wha t he was to ca l l in 1909, the "poin t de rue g6om6t r ique" .
I t i s unf or tun a te tha t BOREL did n ot re l a te th i s examp le in the pap er of 1905 to
his ideas of 1903 discussed abov e, by conc ludin g fur t her th at th e se t of poin ts c~
sat is fying for a ny f ixed n > 2 the ine qua l i ty
fo r infinitely many d i s t inc t ra t iona l s p/q has measure zero. As a resul t , we lack
d o c u m e n t a r y e v id e n ce o f a "CANTELLI" - l i ke asser t ion by BOREL that
i n a m e a s u r e - t h e o r e t i c i n t e r p r e t a t i o n o f p r o b a b i l i t y b y B O RE L , e v e n i n t h e c a s e o f
g e o m e t r i c p r o b a b i l i t y w h e n h e w a s e x p l i ci tl y a w a r e o f s u c h o n i n t e r p r e ta t i o n .
4 ) A r e m i n d e r t h a t B - m e a s u r a b l e s e t s f o r m , i n c o n t e m p o r a r y t e r m s , a a -
a l g e b r a , a n d t h a t t h e a s s o c i a t e d m e a s u r e i s , a g a i n i n c o n t e m p o r a r y t e r m s , o r -
a d d i t i v e .
I n s u m : B O R E L 's i d e n t i f i c a t i o n o f g e o m e t r i c p r o b a b i l i t y w i t h m e a s u r e is e x p li c i t
i n 1 90 5, a n d f u r t h e r t h e s c o p e o f (g e o m e t r i c ) p r o b a b i l i t y i s e x p l ic i tl y e n l a r g e d , t o
a p p l y t o a a - a l g e b r a o f s e ts ( c f fo ot no te 1 , § 6 .2) .
O u r a n a l y s is o f t h e p a p e r o f 1 90 9 a b o v e s h o w s b y c o n t r a s t t h a t o n l y i n t h e
s p e ci a l a p p l ic a t i o n s ( n a m e l y d e c i m a l e x p a n s i o n s a n d c o n t i n u e d f r a c ti o n s w h e r e t h e
c o n v e n t i o n i s e x p l ic i t ly m a d e t h a t t h e p r o b a b i l i t y o f a n i n t e r v a l i s i ts l e n g t h ) is it
e v e n m a r g i n a l l y p o s s ib l e t h a t B O R E L h a d t h e s a m e f a c ts i n v ie w . In t h e p a p e r o f
1 9 0 9 t h e s e e x a m p l e s a r e c o n s i d e r e d o n l y a f t e r t h e f u n d a m e n t a l Z e r o - O n e L a wc o n c e r n i n g A ~ i s o b t a i n e d . T h i s r e s u l t is in t u r n c o n s i d e r e d i n d e p e n d e n t l y o f
g e o m e t r i c c o n s i d e r a t i o n s a n d i n t h e s e e m i n g c o n v i c t i o n t h a t c o u n t a b l e i n d e -
p e n d e n c e i s th e k e y n o t i o n , c a s t i n g s u b - a d d i t i v i t y a n d o - - ad d i ti v it y a s i de . F o r
B O RE L , t h e " p o i n t d e v u e l o g i q u e " h a d e c li p se d t h e " p o i n t d e v u e g 6 o m 6 t r i q u e " i n
1 90 9 ev e n t h o u g h b o t h w e r e a p p l i c a b l e a n d i n s p it e o f t h e a d d i t i o n a l i n s ig h t s
( r e c o g n i z e d i n 1 9 0 5 ) o f f e r e d b y t h e l a t te r .
8 . The Bore l -Berns te in Polemic
8 . 1 . I n t r o d u c t i o n
B O R EL 'S r e s u l t s c o n c e r n i n g d e c i m a l ( o r d y a d i c ) d ig i t s a n d c o n t i n u e d f r a c t i o n s
a t t r a c t e d c o n s i d e r a b l e s t i r i n t h e y e a r s i m m e d i a t e l y f o l lo w i n g 1 9 0 9 . I n 1 91 0 FA B E R
p r o v e d a g a i n t h e r e s u l t c o n c e r n i n g t h e d e c i m a l d i g it s, t h o u g h i n a s u b s t a n t i a l l y
d i f f e r e n t w a y ( c f §9 .2 ). In 191 1 F . BERNSTEIN a t t a ck ed BOREL's p ro o f o f the
C o n t i n u e d F r a c t i o n T h e o r e m , s u p p l y i n g a n a l t e r n a t i v e o f h i s o w n . B O R E L
r e s p o n d e d t o B E R N S TE IN 's p a p e r w i t h a m o d i f i c a t i o n o f h is Z e r o - O n e L a w ( 1 91 2),
a d d i n g t h a t t h i s m o d i f i c a t i o n , c o u p l e d w i t h i n e q u a l i t i e s a l r e a d y i n t h e p a p e r o f
1 90 9, s u ff i c ed t o v a l i d a t e h i s r e s u l t o n c o n t i n u e d f r a c t i o n s . T h e e x c h a n g e w i t h
BERNSTEIN of fe rs a n o p p o r t u n i t y t o e x a m i n e o n c e a g a i n B O R E L ' s o w n i n -
t e r p r e t a t i o n o f " p r o b a b i l it 6 s d 6 n o m b r a b l e s " . I n p a r t i c u l a r, i t d ec i si v e ly s u st a in s
o u r a s s e r t io n t h a t " c o m p o s i t e p r o b a b i l i t y " l a y a t t h e h e a r t o f h i s t h e o r y , a n d t h a t
c o u n t a b l e a d d i t i v i t y ( a n d i t s c o r o l l a ry , c o u n t a b l e s u b - a d d i t i v i t y fo r n o n - d i s j o i n t
u n i o n s ) w a s i n n o w a y c e n t r a l t o B O R EL 's c o n c e p t i o n o f p r o b a b i l i t y . B O R E L (1 91 2)
a l so i n d i c a te s h o w f a r BO R EL w a s f r o m a p p l y i n g " C A N T E L L I " r e a s o n i n g t o t h e c a s e
~ p , c o n v e r g e n t : i n d e e d , t o s h o w h i s (1 91 2) m o d i f i c a t i o n o f h is Z e r o - O n e L a w
v a l i d a t e s t h e C o n t i n u e d F r a c t i o n T h e o r e m ( b o t h c a s e s ) , h e c h o s e t h e c o n v e r g e n t
c a se f o r d e t a i l e d e x p o s i ti o n , a l t h o u g h t h e " C A N T E L L I " r e a s o n i n g (i.e. u s e o f s ub -
a d d i t i v i t y ) p r o v i d e s a s t r o n g e r a n d s i m p l e r m o d i f i c a t i o n fo r th i s c as e .
F i n a l l y , h e m a r r e d h i s o w n d e f e n s e b y i n s i s t in g t h a t a s p e c if ic in e q u a l i t y o f
BOREL (1909) w as e qu iva le n t to one o f BERNSTEIN 'S (1911) . In f ac t wh i l e the de s i re d
i n e q u a l i t y w a s p e r h a p s l a t e n t i n t h e r e a s o n i n g o f BO R E L 's p a p e r o f 1 9 0 9 , o n l y a
h o p e l e s s l y w e a k e n e d v e r s i o n w a s e x p l i c i t l y g i v e n i n t h e p a s s a g e c i t e d b y B O R E L .
BERNSTEIN'S paper is devoted to inves t igat ing the m e a s u r e of the poi nts (p, {) in the
uni t sq uare for which a mea n mo t io n ex is ts . His r e su l t is tha t th i s se t i s o f me asu re
zero.
The s ign i f icance of th i s f o r Ce les ti a l M echa nics i s no t o f in te r es t he r e .
BERNSTEIN shows how t he r e s u lt r e duc e s t o de t e r m i n i ng t he m e a s u r e o f po i n t s i n
t he un i t in t e r va l wh i c h pe r m i t a p p r o x i m a t i o n t o h i gh de g re e by m e a n s o f r a t i ona l
app rox im ant s . Because of th is , BERNSTEIN i s l ed to d i scuss the m easu re of
( i rr a t iona l ) num be r s i n t he un i t i n t e r va l whos e c on t i nue d f r a c t ion e xpa ns i ons ha ve
inf in i te ly m an y e lem ents an which a r e l a rge . Spec i fica lly , he es tab l i shes and em plo ys
t he r e s u lt t ha t t he s e t o f x who s e c o r r e s pond i ng e l e m e n t s a , = a , ( x ) are un-
bou nde d a s n r a nge s ov e r the od d i n t e ge r s is o f m e a s u r e 1.
C onve n i e n t l y , a ll t he t he o r e m s a b ou t c on t i nue d f r a c ti ons t ha t he e s ta b l i she s a r e
g r ou pe d i n a s e l f -c on t a i ne d s e c ti on e n t i t le d " T he ge om e t r i c p r oba b i l i t y f o r t he
a p p r o x i m a t i o n o f r e al n u m b e r s b y r a t i o n a l n u m b e r s , t o s t ro n g e r o r d e r t h a n
c on t i nue d f r a c t ion a pp r o x i m a t i ons , a nd r e l a t e d t op i c s " . T he c h i e f r e s u lt e m p l o ye d
f o r a p p l i c a ti o n s o f m e a n m o t i o n s is hi s T h e o r e m 2 :
Those i rrat ional numbers x in (0, 1) which sat i s fy
a~ <k o r a n r ~ k , k > l
f o r r= 0 , 1, 2 , . . . along so me spec i f i ed subsequ ence n 1 < n 2 < n 3 < . . . < n ~ . .. are
poin t s o f measure zero . BERNSTEIN (1911: 428).
Ta kin g the un io n of these "N ul l -m en ge " for k - - 1, 2 , 3 . . . c lea r ly impl ies the r esu l t
ab ou t un bo un de d gro wt h of a , wi th od d ind ices c i t ed above . BERNSTEIN fur the r
e s t a b l is he s t he B OR EL C on t i nue d F r a c t i o n T h e o r e m i n a ne w manner (BERNSTEIN
( 1911: 256) T he o r e m 4). H i s p r o o f is une xc e p t i ona b l e , a l t houg h h is s um m i ng up o f
t he a r gu m e n t i n the f o r m o f a t he o r e m i s c l oudy . He i s a t pa i n s t o po i n t t o t he f l a wed
cha rac te r o f BOREL' s pr oo f in i ts use of inde pend ence be tw een " t r i a l s " a ,__>q~(n),
a nd ne ve r t o u s e s uc h r e a s on i ng h i m s e l f .
We n ow ske tch BERNSTEIN's sec t ion 2 . To the n o ta t io n a l r eady in t roduc ed , we
a d d t h e n o t a t i o n
P [ a ,> = k l a l = m i , a 2 = m 2 . . . . . a , _ l = m , _ l ]
t o de no t e t he condi t i ona l probab i l i ty t ha t a , ( x ) > k g i ve n t ha t ai (x ) = m~, i = 1, 2 .. .. ,n - 1. This i s, by de f in i t ion , a r a t io of l engths : the nu m era to r i s
l [ al = m i , . .. , a , - l = m , - 1 , a , = m , ] ,m n ~ k
t h e " l e n g t h " (i.e. m e a s u r e ) o f t he po i n t - s e t
[a I =m 1, . . . , an_ 1 = mn_ 1, a , = m n ]m n ~ k
a n d t h e d e n o m i n a t o r i s
l [ a i = m l . . . . . a , l = m , - i ] ,
the l ength of a sing le in te rva l co r r esp ond ing to the se t in which a i ( x ) = m i ,
(1 k 2 + l ) . . . ( 1 - k - - ~ ) < P [ a , l < k . . . . . , a , r < k . r ]r
(8.6)
T h e s e b o u n d s a r e f o r " g l o b a l ' , n o t c o n d i t i o n a l p r o b a b il i t ie s . M o r e i m p o r t a n t , t h e y
a r e f o r t h e p r o b a b i l i t i e s o f a p a r t i c u l a r k i n d o f c y l i n d e r s et , n a m e l y i n t e r s e c t i o n s o f
t h e i n d i v i d u a l s e ts (o r " t ri a l s " ) d e f i n e d b y i n d i v i d u a l i n e q u a l i ti e s o f t h e f o r m a , > k ,
( o r a , < k , , r e s p e c t i v e l y ) , fo r v a r i o u s f i n it e c h o i c e s o f n . M o s t i m p o r t a n t o f a ll , t h e s e
i n e q u a l i t i e s c o i n c i d e w i t h w h a t o n e c o u l d h a v e o b t a i n e d f r o m t h e i n d i v i d u a l
" g l o b a l " i n e q u a l i t i e s
1 2~ < P [ a , > = k , J < k , + l
a n d
2 11 - < P [ a , < k , ] < l - - -
k , + 1 k ,
( e s se n t ia l ly e q u i v a l e n t t o t h e " g l o b a l " i n e q u a l i ti e s o f B O R E L ; c f (8 .3) , an d (8 .4)) h a d
o n e b e e n a l l o w e d t o a s s u m e i n d e p e n d e n c e a s w e l l .
B E R N ST E IN 'S i n g e n i o u s a r g u m e n t f o l lo w s f r o m t h e p u r e l y a l g e b r a i c i d e n t i t y
P [ a i = m a , a2 ~m2 , . . . , an =toni
= P [ a , = m , l a i= m i , l _ < i < n - 1 ] P [ a , _ i = m , _ i j a i = m i , l _ < i < n - 2 ] (8.7)
• . P [ a 3 = m 3 1 a l = m i , a z = m 2 ] P [ a2 = m z l a i = m l ] P [ a l = m l J .
T h i s i s t h e f o r m o f " p r o b a b i l i t 6 s c o m p o s 6 e s " w h i c h is a d e q u a t e l y g e n e r a l f o r t h e
c a s e a t h a n d . T h i s m i g h t w e l l b e c a l l e d t h e ( f i n i t e ) C h a i n L a w o f P r o b a b i l i t y .
T o p r o v e B E R N S T E I N 'S i n e q u a l i t i e s ( 8 .5 ) a n d (8 .6 ) f r o m t h e C h a i n L a w , d e n o t e
t h e p r o d u c t o n t h e ri g h t o f t h e C h a i n L a w b y I ~ ( m i , m 2 . . . . . m , ). F i xtl
m l , m 2 , . . . , m , _ 1 a n d s u m b o t h s id e s fo r al l v a l u e s o f r e, r a n g i n g f r o m a f i x ed l o w e r
l i m i t t o i n f in i ty . I f t h e l o w e r l i m i t i s 1, b o t h s i d e s s im p l i f y , a n d t h e r e s u l t i s t h eo r i g in a l a s s e r t io n w i t h a l l r e f e re n c e s t o a , d e l e te d , i n v o l v in g I ~ o n t h e ri g h t - h a n d
n-1
s id e . I f t h e l o w e r l i m i t i s s o m e i n t e g e r k , > 1, t h e l e f t - h a n d s i d e b e c o m e s
P [ a l = m l , . .. , a , _ i = m , _ l , a , > k , ]
w h i l e th e r i g h t - h a n d s i d e o f (8 . 7 ) c a n b e e s t i m a t e d f r o m a b o v e a n d f r o m b e l o w a s t h e
p r o d u c t
I-[ = P [ a , _ i = m , _ i I a i = m i , 1 < - - i< _ n - 2 ] . .. P [ a 2 = m z l a i = m i ] P [ a i = m a ]n- - i
t i m e s t h e u p p e r a n d l o w e r e s t im a t e s , re s p e c t iv e l y , w h i c h h o l d f o r th e l e a d i n g f a c t o r
1 2k , , -1 [ I < P [ a i= m i , l<- i<-n- l,a ,>k ,]<--= k , + l , - 1 1 ~ '
I n a s i m i l a r f a s h i o n w e e s t i m a t e [ I f r o m a b o v e a n d b e lo w , o r r e d u c e i t t o ~ I ,n - - 1 n - - 2
a s fo l l o w s : W e s u m o v e r m , _ 1 f r o m s o m e l o w e r l i m i t t o in f in i ty , d e l e t in g r e f e r e n c e
t o a , 1 i f t h i s l o w e r l i m i t is 1, o r e l s e o b t a i n i n g
1 2k° f I < I I < f I- - n - - 2 n - - 1 k n 1 + 1- n - - 2
i f t h e l o w e r l i m i t i s k , _ 1 > 1. P r o c e e d i n g i n t h is f a s h i o n , u s i n g a p r e c r i b e d f in i te
c o l l e c t i o n o f i n d i c e s n 1 . . . . n r w i t h p r e s c r i b e d l o w e r l i m i t s k ,~ , k ,2 . . . . , k , r , w e a r r i v ea t t h e d e s i r e d i n e q u a l i t y (8 .5 ). T h e m i s s in g i n t e r m e d i a t e i n d ic e s c o r r e s p o n d t o
s u m m a t i o n s w i t h l o w e r l i m i t 1. T h e i n e q u a l i t y (8 .6 ) f o r P [ a , ~ < k . . . . . . a , r < k j i s
o b t a i n e d s i m i l a r l y or , a l t e r n a t i v e l y , c a n b e v i e w e d a s a n i m m e d i a t e c o r o l l a ry .
B E R N S T E IN n e x t e x p l i c it ly e m p l o y s t h e c o n c ep t o f m e a s u r e a n d t h e va r i ou s f o r m s
o f c o u n t a b l e a d d i t i v i t y ( r e f e rr i n g t o L E B E S G U E ( 19 0 6) ) i n o r d e r t o e x t e n d p r e v i o u s
c o n s i d e r a t i o n s t o s e t s d e f i n e d b y i n fi n it e ly m a n y i n e q u a l it ie s . H e t h u s c a l c u l a te s
t h e p r o b a b i l i t y o f t h e s e t
[ a ,~ > k , 1 , a , 2 > k , ~ , . . . , a , > k , ~ , . . . ]
a n d o f t h e s e t
[ a,~ < k n~ , a , ~ < k , ~ , . . ., a , < k . . . . . . ] .
T h e s e s e ts i n v o l v e in f i n it e s e q u e n c e s o f i n e q u a l i t i e s a n d t h u s a r e n o l o n g e r c y l i n d e r
s e ts . F o r i n s t a n c e , h i s T h e o r e m (2 ), r e f e r r e d t o a b o v e , r e s u l ts b y a p p l y i n g
B E R N S T E I N ' s i n e q u a l i t i e s ( 8. 5) a n d ( 8 .6 ) t o a g i v e n s e q u e n c e { n, } o f i n d i c e s , a n d
c h o o s i n g k ,~ = k f o r r = 1 , 2 , 3 , . . . . S i n c e
r . r
l i m ( l _ k ~ ) = l l m ( 1 _ ~ ) ~ = l i m ( ~ )r = l i m 1
i n e q u a l i t i e s ( 8.5 ) a n d ( 8.6 ) a n d c o u n t a b l e a d d i t i v i t y i m p l y t h e t w o r e s u l t s :
P [ a , i > k , a , 2 > k . . . . . a , r > k , . .. ]
= l i m P [ a , 1 > k , a ,2 > =k , . . . , a , > k ] = 0r~oo
a n d
P [ a , 1 < k , a ,2 < k , . . ., a , . < k, . . . ] = l i m P [ a , 1 < k , . . . , a , . < k ] = 0 .r ~ c o
T h e s e t s [ a , . > k , r = 1 , 2 , 3 . . . ] a n d [ a , r < k , r = 1 , 2 , 3 , . . . ] a r e b o t h o f m e a s u r e z e r o
f o r e a c h v a l u e o f k . H e n c e u n i o n s o f s e ts s u c h a s k = 2 , 3 , . . . a r e s t il l o f m e a s u r e z e r o ;
t h u s t h e p r o b a b i l i t y t h a t { a ,r } is b o u n d e d f r o m a b o v e ( o r b e l o w ) i s z e r o .
T o o b t a i n B O R E L'S t h e o r e m o n C o n t i n u e d F r a c t i o n s , i t s u ff ic e s t o a p p l y ( 8.5 )
a n d ( 8 .6 ) t o t h e i n d i c e s n , n + 1 . . . , n + r , a n d c h a n g e k , , k , + 1 . . . , k , + r t o t h e B O R E L
n o ta ti o n qS(n), qS(n+ 1 ), . . . , qS(n + r). Th en fro m ine qu ali t y (8.6) i t follows t ha t
1 ¢ (v ) -+ l < P [ a v < ( ~ ( v ) ' v = n ' n + l . . . . . r ] < I ] - •v = n v ~ n
c~
If ~ 11 0 (~ d i ve rge s ,
P [ a ~ < O ( v ) , v = n , n + 1, . . . ] = l im Pl-a~ < qS(v), v = n , n + 1 . . . . r] = 0r ~ 3
1(the diver gent case of BOREL's theor em) . I f ~ ~ conve rges , (8.5) assures that
P[a~ < O(v) , v= n , n + 1, . . . , r]
i s pos i t ive (and less than 1) and in fact l ies between
( ) a n d f i 1 - ~fi 1 q~(v)+l-
~ n v = n
To achie ve BOREL's resul t in this case , i t i s nece ssary to co m pu te the p rob ab i l i ty
tha t a~ < ~b(v) hold " f r om some n on" , tha t i s the pro babi l i ty of the unio n of the se ts
[av< ~b(v ) ,v> n] over a l l in teger va lues of n . Thi s pro babi l i ty i s thus 1, aga in
va l ida t ing BOREL's resu l t. I f the phrase " f r om some n on" is in te rp re ted to me an" f r o m s o m e f i x e d v a l u e of n on " then the answer i s s t r ic t ly l es s than one . Thi s
amb igui ty o f l anguage resu l t ed in needles s , and for our purpose , i r re l evant conf l i c t
be twe en w ha t BOREL sa id he pr ov ed a nd wha t BERNSTEIN ac tua l ly succeede d in
proving. Their resul ts coincide, but BERNSTEIN actual ly suppl ied a val id,
i nde pe nde n t l y c onc e i ve d p roo f , e xp li c it ly u ti li z ing t he l a ngua ge a nd t he o re ms o f
mea sure theory . BOREL, by cont ras t , f ir s t a s se r t ed the th eor em in 1909 but the re
s upp l i e d a non -p roo f , v i t i a te d by de pe nde nc e be t w e e n " t r i a l s " a nd e mpl oy i ng on l y
t he Z e ro -O ne L a w . A s w e ha ve s e e n , t he Z e ro -O ne L a w p rove d by B O R E L ha s
noth ing to d o wi th the l angu age of LEBESGUE measure , and , in B O R E L ' S expos i t ion ,
has no c lea r ly recog nized l ink to co unta ble addi t iv i ty or sub-addi t iv i ty .Af te r achieving h i s pr oo f of BOREL'S resu l t on Con t inu ed Frac t ions , BERN-
STEIN emphas izes tha t the indiv idua l event s a n = m . a re no t i nde pe nde n t . H e
c ompu t e s t he fou r p roba b i l i t i e s P [ a 1 = 1], P [ a 1 = 2 ] , P [ a I = 1, a 2 = 1], P [ a 1 = 2 ,
a 2 = 1] as lengths of intervals , and obser ves tha t the tw o ra t ios
P [ a 1 = 1, a 2 = 1 ] a n d P [ a l = 1]
P [ a l = 2 , a 2 = 1 ] P [ a 1 = 2 ]
a r e une qua l , a s t r ue i nde pe nde nc e w ou l d r e qu i r e . H e po i n t s ou t t ha t t he a de qua t e
law of "probabi l i t6s com pos6 es" (or a s BERNSTEIN ca ll s it "da s Th eo rem der
z us a mm e nge s e t z t e n W a hr s c he i n l i c hke i t e n" ) invo l ves p rodu c t s o f compos i t e p r o b a -
b i li ti e s to co mp ute p robabi l i t i e s o f the s imul tan eous occu r rence of severa l events .
Thi s i s wha t we have ca l l ed the Cha in Law. The example he g ives , in modern
nota t io n , cons ide rs two event s , A and B , where A - - • Ai deco mp oses event A in to
i ts poss ible a l ternat ives . Then
P(A c~ B) = P (@ (A i c~ B)) = ~ P(A i) P( B ]Ai).
I f q < P (B I Ai) < ~1for all i , then
q P(A ) ~ P(A ~ B) __<qP(A) .
Thus , bo und s on c ondi t io na l prob abi l i t i e s for B , va l id for a ll a l t e rna t ive condi t ions ,
g ive r i se to the same inequ a l i ty concern ing P(A ~ B) as wou ld hav e been ob ta ined i f
the g loba l probabi l i ty P (B) were known only to sa t i s fy
q< P(B)<~I
and the event s A and B were independent .
Thi s obse rva t ion could have been appl i ed d i rec t ly to BOREL's Zero-One Law
i n t he de g re e o f ge ne ral i ty o f " de nu me r a b l e p roba b i l i t y " . S inc e, how e ve r ,
BERNSTEIN'S w hol e v i e w po i n t is no t t o c r e a t e a ne w t he o ry o f de num e ra b l e
p roba b i l i t y bu t t o e m pl oy t he e x i st ing t he o ry o f me a s u re , he doe s no t m a ke t he
observa t ion tha t we can make on h i s beha l f : BERNSTEIN in t roduced the key
e lement in genera l i z ing the BOREL Zero-One Law in the case not cons ide red by
CANTELLI, i.e. the d ive rgent case . CANTELLI dem oted the h ypothes i s of inde-
pendence in BOREL's Zero-One Law in the convergent case ( the "ze ro" ha l f ) by
showing i t could be omit ted. BERNSTEIN general ized the case of divergence ( the" o ne " ha l l) by s how i ng t he hypo t he s i s o f i nde pe nde nc e c ou l d be w e a ke ne d t o
requi r ing adequa te (upper ) bounds on ce r t a in condi t iona l probabi l i t i e s , these
bounds themselves forming a divergent ser ies . In fact , had he been suff ic ient ly
thoro ugh -go ing in h is use of measu re the ory , and had he no t ins i st ed , like BOREL,
on t rea t ing the conv ergen t and d ive rgent cases in like man ner , he cou ld have show n
tha t in the conv ergen t case sub-addi t iv i ty es t imates a lone w ould h ave es tab l i shed
the theo rem in te rms of the g loba l probabi l i t i e s P[an > ~b(n)]. He wou ld thus hav e
ant ic ipated CANTELLI. Indeed, in his sect ion (1) , he expl ic i t ly calcula tes the
mea sure of the l imi t in fe r ior of a co l l ec t ion of set s, a t t r ibu t ing th is ca lcu la t ion to
LEBESGUE (1906). This is , in fact , a m od er n "CANTE LLI-l ike" calcula t ion . Ho we ve rthat may be, the fact i s that BERNSTEIN gave the first valid proof of BOREL'S
result on Continued Fractions, and gave the first generalization of BOREL'S Zero-
One La w to cover the possibility of dependence.
8.3. Borel's Response: Borel (1912)
After BERNSTEIN'spaper , the s t a tus of BOREL'S Con t inue d Fra c t io n T he ore m
was, for a short per iod, in doubt . Both BERNSTEIN and BOREL agreed that the
prob abi l i ty tha t a , < qS(n) should hold f rom some n on was ze ro i f ~ d ive rged .
BOREL asser ted this in the form that the p rob abi l i ty t hat a , > qS(n) hold s inf ini te ly
1often, the Aoo of his pa pe r of 1909, is 1. In the case ~ ~ con ve rge nt, BOREL
a s s e r t e d t h a t t h e p r o b a b i l i t y t h a t a n > ~ b ( n ) h o l d s i n f i n i t e l y o f t e n i s z e r o ; t h u s
a n < ~b(n) h o l d s f r o m s o m e n o n w i t h p r o b a b i l i t y o n e . B E R N S T E I N a s s e r t e d i n t h e
c o n v e r g e n t c a se t h e p r o b a b i l i t y t h a t a n < g ) ( n ) h o l d s f r o m s o m e n o n ( m e a n i n g
a n < ~ b(n ), a n + 1 < ~ b( n + 1 ), . . . f o r a g iv e n n ) i s p o s i t i v e b u t l e s s t h a n o n e . BE R N S T E I Nt h u s c r i ti c i z ed b o t h B O R E L'S a r g u m e n t a n d h i s r e su l t. S t u n g b y t h is , B O R E L
r e s p o n d e d w i t h a n o t e (B O RN E ( 19 1 2) ). T h e r e h e i n s i s te d t h a t h i s r e s u l t w a s c o r r e c t
a s s t at e d , n a m e l y A ~ is z e r o o r o n e i n t h e c o n t i n u e d f r a c t i o n c a s e a c c o r d i n g a s
1~ c o n v e r g e s o r d i v e rg e s , b u t c o m p l e t e l y r e w o r k e d th e p r o o f i n s u c h a f a s h i o n
t h a t d e p e n d e n c i e s w e r e pe r m i t te d . H a v i n g g e n e r al i z ed h i s o r i g in a l Z e r o - O n e L a w
i n t h is w a y , B O R E L a t t e m p t e d t o a p p l y i t t o p r o v e h i s c o n t i n u e d f r a c t i o n r es u lt . A s
w e s h a ll s ee , t h e a p p l i c a t i o n r e q u i r e d a m o d i f i c a t i o n o f h i s o ri g i n a l c a l c u l a t i o n s o f
1 90 9. T h i s B O R E L f a il e d to u n d e r t a k e , p e r h a p s u n w i l l i n g t o c o n c e d e s u c h a n
i n a d e q u a c y i n h i s c a l c u l a t io n s o f 1 90 9.
B O R E L ' s n e w p o o f o f t h e g e n e r a l i z e d Z e r o - O n e L a w e s s e n t i a ll y c o i n c i d e s w i t h
t h e a r g u m e n t o f B E RN S TE IN . T h a t is, th e g e n e r a l l a w o f " p r o b a b i l i t ~ s c o m p o s 6 e s "
( in t e r m s o f c o n d i t i o n a l p r o b a b i li t ie s ) , r e p l a c e s c o u n t a b l e i n d e p e n d e n c e a s t h e
e s s e n ti a l to o l . B O R E L t h e n s h o w s t h a t a p p r o p i a t e i n e q u a l it i e s o n c o n d i t i o n a l
p r o b a b i l i t i e s g i v e r is e t o " i n d e p e n d e n c e - l i k e " e s t i m a t e s i n th e f o r m o f p r o d u c t s (cf.
(8 .5 ), (8 .6 ), a b o v e ) . B u t B O R E L , i n 1 9 1 2 , p l a c e s t h e s e a r g u m e n t s i n t h e g e n e r a l i t y o f
h i s o r i g i n a l Z e r o - O n e L a w , i.e., t h e s p a c e o f a ll d e n u m e r a b l e s e q u e n c e s o f t ri a ls ,
w i t h p o s s i b l e s u c c e s s o r f a i l u r e a t e a c h t ri a l, w h e r e a s B E R N S T E I N h a d b e e n
c o n c e r n e d o n l y w i t h t h e a p p l i c a t i o n t o c o n t i n u e d f r a ct i o ns .
B O R E L ' s o r i g i n a l n o t a t i o n Pn f o r s u c c e s s a n d qn f o r f a i l u r e a t t h e n th t r i a l i s
i n a d e q u a t e f o r h is n e w p r o o f. W e a l s o n e e d t h e c o r r e s p o n d i n g c o n d i t i o n a l
p r o b a b i l i ti e s , s i nc e w e e x p l i ci t ly a l lo w d e p e n d e n c e . L e t u s i n t r o d u c e t h e n o t a t i o n
s , f o r a " s u c c e s s " p a r a m e t e r : s , = 1 m e a n s t h e n th t r i a l w a s a s u c c e s s, s n = 0 m e a n s
i t w a s a f a i lu r e . L e t x b e a n i n f in i t e s e q u e n c e o f t r i a l s ; t h e n s n ( x ) i s 1 i f x ha s
s u c c e s s a t t h e n th t r i a l a n d 0 i f x h a s f a i l u r e a t t h e n th t r i a l. T h e p r o b a b i l i t y t h a t a
s e q u e n c e h a s p r e s c r i b e d i n i t i a l v a l u e s s i = m l , i = 1 , 2 . . . . . n w h e r e e a c h m i = 0 o r 1 i s
g i v e n r e c u r s i v e l y b y
P E s 1 = m 1 . . . . s n = m , ] = P [ s , = m n Is 1 - - - - m I . . . . . S n - 1 = r a n _ i ]
. P [ S l = m l , . . . , s n _ l = m , _ l ]
s o t h a t"_L
P [ s 1 = m 1 . . . . , s , = r n , ] = I I P [ s k = r n k I s1 = / T / l , " ' , S k - 1 : l T l k - - 1 ] "
k = l
T h i s i s p r e c is e l y th e C h a i n L a w . S i m i l a r l y t h e p r o b a b i l i t y t h a t a s e q u e n c e f in i sh
w i t h t h e r e s u l t s S n + l = m n + p S n + 2 = m n + 2 . . . . g i v e n t h e i n i t i a l v a l u e s s l = m l , . . . ,
s , = m , is t h e i n f i n i t e p r o d u c t
f i P [ S , + k = m , + k l S l = r n l . . . . . s , + k - 1 = m , + k - 1 ]. (8 .8)k = l
T h i s i s t h e c o u n t a b l e e x t e n s i o n o f t h e C h a i n L a w t h a t B O R E L a s s er t s i n 1 91 2.
( B O R E L o f f er s n o j u s t i f i c a t i o n f o r t h e e x t e n s i o n f r o m t h e f in i t e t o t h e c o u n t a b l e c a s e ,
o n e m o r e i n s t a n c e o f a t a c it a n d e x t r e m e l y w e ll d i sg u i s ed d e p e n d e n c e o n c o u n t a b l e
a d d i t i v i t y . ) I t i s e m p l o y e d i n h i s r e s p o n s e t o B E R N S T E I N t o p r o v e t h e f o l l o w i n g
t h e o r e m : L e t A o , A s . . . . . A k . . . . b e , a s in 1 90 9, t h e p r o b a b i l i t y t h a t t h e r e a r e e x a c t l y
k s u c c e ss e s , a n d l e t A o~ = 1 - ( A o + A s + . . . + A k + . . . ) . L e t { p',} a n d {p', '} b e t w os e q u e n c e s s u c h t h a t
p ' , < P [ s n = l l s s = m l . . . . . S , _ l = m ~ _ s] < p~ '
f o r m l , m 2 , . . ., m , _ s r u n n i n g t h r o u g h a l l c h o i c e s o f 0 ' s a n d l 's .
I f ~ p " c o n v e rg e s , t h e n A , = 0 .
I f ~ p ' , d i v e rg e s , t h e n A , = 1.
E q u i v a l e n t l y ,
I f V p " c o n v e r g e s, t h e n A o + A 1 + . . . + A k + . . . . 1., n
I f ~P'n d i v e r g e s , t h e n A o + A I + . . . + A k + . . . . O.
B O RE L p r e s e n t s a p r o o f o n l y f o r t h e c o n v e r g e n t c a s e, s in c e t h a t is th e c a s e f o r
w h i c h B E R N S T EI N th i n k s ( e r r o n e o u s l y ) h i s r e s u l t i s i n c o n t r a d i c t i o n w i t h B O R E L 'S .
O f c o u r se "CANTELLI" r e a s o n i n g r e n d e r s BOREL's n e w p r o o f o f t h i s c a s e o b s o l e t e ,
b u t B O R E L 's p r o o f i n d i c a te s h o w t o h a n d l e t h e d i v e r g e n t c a s e a s w el l, a n d t h e r e f o r e
m e r i t s a t t e n t i o n .
I f ~ p ~ ' c o n v e r g e s , c o n s i d e r A o + A s + " " + A k . B O R EL s h o w s t h a t g i v e n e > 0 , k
c a n b e c h o s e n s o l a r g e t h a t
A o + . . . + A k > 1 - ~ .
T h e p r o o f is a s tr a i g h t f o r w a r d a p p l i c a t i o n o f t h e g e n e r a li z e d l a w o f c o m p o s i t e
p r o b a b i l i t i e s (8 .7 ): A 0 + . .. + A k is t h e p r o b a b i l i t y o f h a v i n g a t m o s t k s u c ce s s es ,
t h e r e f o r e g r e a t e r t h a n t h e p r o b a b i l i t y o f h a v i n g n o s u cc e ss e s f ro m t h e ( k + 1) st on . In
s y m b o l i c f o r m
A 0 + ' " + A k > P [ S k + 1 = 0 , Sk+ 2 = 0 . . . . ] .
B u t t h e l a t t e r is t h e w e i g h t e d a v e r a g e o f t h e p r o b a b i l i t i e s
P[Sk+ 1 = 0 , Sk+2 = 0 , . .. I S1 = m s . . . . , S k = m k ]
( cf. (8 .8 )) f o r a l l 2 k c h o i c e s o f ( m l . . . . , m k ). B y u s e o f t h e c o u n t a b l e e x t e n s i o n o f t h e
C h a i n L a w e a c h o f t h e s e p r o b a b i l i t i e s ( a n d h e n c e t h e i r w e i g h t e d a v e r a g e P [ sk + 1
- - 0 , S k ÷ 2 = 0 . . . . ] ) is e s t i m a t e d f r o m b e l o w b y t h e i n fi ni te p r o d u c t
( 1 - - p 'k '+ l ) ( 1 - - p 'k '+ 2 ) . . . I ~ ( 1 - - p ) ' ) .j = k + l
I f ~ p y < o% t h e n k c a n b e c h o s e n l a r g e e n o u g h t o m a k e t h is p r o d u c t g r e a t e r t h a n
S i m i l a r l y , i f ~ p j d i v e r g e s , A o + A I + . . . + A k c a n b e s h o w n t o v a n i s h , t h o u g h
B O R EL o m i t s t h is . A p r o o f c a n b e b a s e d o n t h e i n e q u a l i t y
P I s ,+ 1 = 0 , s , + 2= 0 , . . . , S N = 0 I S 1 = m l , . . . , S n = t o n i < ( 1 - p ~ ,+ 1 ). .. (1 - p } ) ,
s u p p l e m e n t e d b y c o n t i n u i t y (i.e. c o u n t a b l e a d d i t i v i t y ) ; cf. BARONE (1974) .
I n s u m m a r y , B O R EL i n 1 9 12 p r e s e n t s a n e w , m o r e g e n e r a l Z e r o - O n e L a w , b a s e d
o n a n e w , m o r e g e n e r a l m e t h o d o f p r o o f ( b a s e d o n t h e c o u n t a b l e c h a in L a w u t il iz e d
p r e v i o u s l y a n d e x p l i c it ly b y B E RN S TE IN ). C o u n t a b l e a d d i t i v i ty a n d s u b - a d d i t i v i ty
a r e s ti ll n e g l e c te d p r i n c i p l e s ; c o u n t a b l e c o m p o s i t e p r o b a b i l i t y is e m p l o y e d i n th e
f o r m s
oo
r~ o q r n l ~o
. . . . .L ' I ' J L ' I ' J ~ = 5
W e r e p e a t , th e e x t e n s i o n o f t h e C h a i n L a w f r o m t h e f i n it e t o t h e c o u n t a b l y i n f i n it e
c a s e i s o f f e r ed w i t h o u t e x p l a n a t i o n . B y c o n t r a s t , B E R N ST EIN u se s c o u n t a b l e
a d d i t i v i ty e x p l ic i tl y t o c a l c u l a t e t h e p r o b a b i l it i e s o f n o n - c y l i n d e r s et s o f th e f o r moo
( ~ E k , u t i li z i n g t h e a d d i t i o n a l i n s i g h t t h a t t h e s e p r o b a b i l i t i e s a r e ( in e v e r y c a s e1
u n d e r c o n s i d e r a t i o n ) m e a s u r e s o f m e a s u r a b l e s e ts .BOREL f ai ls to r e m a r k t h a t BERNSTEIN'sp r o o f is m u c h t h e s a m e a s t h e o n e h e
n o w p r e s en t s t o p r o v e t h e g e n e r a l iz e d Z e r o - O n e L a w , b u t a s se rt s t h a t " t h e n e w
p r o o f i s e s s e n t i a l l y t h e o n e t h a t w o u l d h a v e b e e n g i v e n i n 1 90 9 if a l l t h e c a l c u l a t i o n s
h a d b e e n w r i t t e n i n f u l l . "
I t o n l y r e m a i n e d f o r B O R E L t o r e a s s u r e t h e r e a d e r ( a n d h i m s e l f ) t h a t t h e
m o d i f i e d Z e r o - O n e L a w a p p li e s to t h e c o n t i n u e d f r a c ti o n ca se .
T h e n e e d e d i n e qu a l it ie s a r e t h e u p p e r a n d l o w e r b o u n d s f o r
P [ a ,> = ¢ ( n) l a l , . . . , a , _ l ]
i n d e p e n d e n t o f w h a t c o n s t r a i n t s a r e p l a c e d o n a l , . . ., a , _ 1 . B E RN S TE IN h a d f o u n d
e x a c t l y s u c h b o u n d s . B O R E L n o w i n 1 9 1 2 a s s e r ts t h a t t h e i n e q u a l i t ie s ( w h i c h h e
r e f er s t o a s " ( 2 3 ) a n d t h e r e a f t e r o n p a g e 2 6 8 " ) o f h i s p a p e r o f 1 90 9 p r o v i d e t h e s a m e
i n f o r m a t i o n . I n f a c t , t h e e s s e n t i a l i n e q u a l i t i e s o f B O R E L i n 1 9 09 a r e , i n t h e o r d e r o f
t h e i r d e r i v a t i o n ,
k P [ a . = k + l ] k + l- - <k + 2 P [ a . = k ] < ~ 3 '
3 k ( k + 1) < P l a n = k ] < ( k + 1 )(k + 2 ) '
2 3< P [ a n > k + l ]3 ( k + 1) = < k + 2 "
N o n e o f t h e s e in v o l v e c o n d i t i o n a l p r o b a b i l it i e s o f t h e s o r t r e q u ir e d . F o r e x a m p l e ,
t he l a s t i s no t
2 33 ( k + 1 ~ < P [ a n > = k + l [ a x = m l . . . . , a n _ l = m , _ l ] < k + 2
a s w o u l d b e r e q u i r e d f o r a p p l i c a t i o n o f t h e g e n e r a l iz e d Z e r o - O n e L a w b u t i s o n l y a
m u c h w e a k e n e d v e r si o n o f it.
B O R E L r e m a i n e d a w a r e t h a t h i s o r i g i n a l e x p o s i t i o n w a s f la w e d , f o r i n 1 9 2 6 h e
p u b l i s h e d a c o n s i d e r a b l y m o r e d e t a i l e d a n d e x p a n d e d v e r s i o n o f h is p a p e r o f 19 09
u n d e r t h e t it le " A p p l i c a t i o n s ~t L ' A r i t h m 0 t i q u e e t/ ~ la T h 6 o r i e d e s F u n c t i o n s " , a s
f a s c ic u l e I o f T o m e I I o f h i s e x t e n s iv e " T r a i t 0 d u C a l c u l d e s P r o b a b i l i t6 s e t d e s es
A p p l i c a t i o n s " . N o w h e p r e s e n t e d t h e m a t e r i a l o f h is r e s p o n s e i n 1 9 1 2 t o
BERNSTEIN (i.e., h i s m o r e g e n e r a l Z e r o - O n e L a w ) b e f o r e t u r n i n g t o C o n t i n u e d
F r a c t i o n s . T h i s t i m e h e p r e s e n t s t h e a b o v e f iv e i n e q u a l i ti e s , n u m b e r e d (1 ), (2 ), (3 ),
(4), (5) , as s t a ted (i .e . , not c o n d i t io n e d ) a n d t h e n r e m a r k s :
T he i neq ua l i t i e s ( 1) , ( 2) , ( 3) , ( 4) , ( 5) r em a i n t r ue i f on e m ak es va r i ou s hy po t he s e s
c o n c e r n i n g t h e e l e m e n t s a l , a 2 . . . . a n 1. T h e s u m o f t h e l e n g th s lk, lk+ 1 o f t h ei n t e rv a l s c o n s i d e r e d , i n s t e a d o f ra n g i n g o v e r a l l p o s s ib l e v a l u e s o f th e e l e m e n t s
a l , . . . , a n _ 1 w i ll o n l y r a n g e o v e r t h o s e v a l u e s s a t i sf y i n g c e r t a i n g i v e n c o n d i t i o n s .
Th e g l ob a l p r o bab i l i t i e s P J an = k ] , P [ a n = k + 1 ] , P [ a , _> k + 1 ] w i l l be r e p l a ced
b y t h e p r o b a b il it ie s o b t a i n e d b y t a k in g a c c o u n t o f t h e h y p o t h e s e s m a d e o n t h e
i n it ia l n - 1 e l em e n t s , a n d w h a t e v e r t h o s e m a y b e , b y f o ll o w i n g t h e s a m e
r e a s o n i n g w h e r e b y t h e y w e r e e s ta b l i sh e d a b o v e t h e p r e c e d i n g i n e q u a l i ti e s w i ll
c o n t i n u e t o b e s a t i s f i e d b y t h e n e w p r o b a b i l i t i e s . B O R E L ( 1 9 2 6 : 6 6 ) .
Th i s s t a t em en t , i n 1926 , i s co r r ec t . BOREL 's a s s e r t i on , i n 1912 , t ha t h i s pape r o f
1909 a l r ea dy co n t a i n ed t he de s i r e d i ne qua l i t i e s i nv o l v i n g P'n an d p~' i s f a l s e o r , a t
l e a s t , d i s i n g e n u o u s .
8 .4 . E a r l y o b s e r v a t io n s o f L e b e s g u e a n d L ~ v y
O n e f in a l h i s to r i c a l c o m m e n t is in o r d e r c o n c e r n i n g B O R E L'S c o n t r i b u t i o n t o
t h e e x c h a n g e w i t h B E R N S TE IN . T h e s e c o n d e d i t i o n o f B O R E L 'S L e f o n s s u r l a T h d o r i e
d e s F u n c t i o n s (B O R E L ( 19 1 4 )) c o n t a i n s m a n y " n o t e s " a d d e d e s p e c i a l l y f o r t h i s
e d i t i o n . N o t e V c o n s i s t s o f a r e p r o d u c t i o n o f B O R E L ( 19 0 9 ) a n d B O R E L (1 91 2), in
to to . A f o o t n o t e w a s a d d e d t o t h is r e p r i n t e d p a p e r o f 1 9 12 w h i c h d o e s n o t a p p e a r i n
t h e o r i g i n a l . I t f o ll o w s t h e s e n t e n c e
W h a t i s v a l i d i n B e r n s t e i n ' s o b j e c t i o n i s t h a t t h e r e a s o n i n g t h a t I h a v e
g i v e n . . . a s s u m e s t h e p r o b a b i l i t i e s a r e i n d e p e n d e n t a n d s h o u l d b e m o d i f i e d
I s h o u l d s a y t h a t t h i s o b j e c t i o n w a s m a d e t o m e i n a p e r s o n a l l e tt e r, b y
L e b e s g u e , a t t h e t i m e o f t h e p u b l i c a t i o n o f t h e Rendiconti. I a s s u r e d m y s e l f t h a t
t h e r es u l ts w e r e v a li d a n d a t t a c h e d n o i m p o r t a n c e t o t h e o b j e c t i o n ; I h a d e v e n
f o r g o t t e n i t w h e n , a f e w y e a r s l a t e r, I r e p l i e d t o B e r n s t e i n : o n l y a f t e r th e
p u b l i c a t i o n o f t h is r e s p o n s e , r e p r o d u c e d h e r e , d i d I c o m e a c r o s s th e o l d l e t t e r o f
L e b e s g u e . B O R E L ( 1 9 1 4 : 2 0 8 : F o o t n o t e ( 4 ) ) .
LEBESGUE and BERNSTEINw e r e n o t a l o n e i n o b s e r v i n g t h a t B O RE L 'S p a p e r o f
1 9 09 s u f fe r e d f r o m d e f e ct s o f e x p o s it i o n . I n a l e t t e r t o u s P . L g v Y w r o t e c o n c e r n i n g
i t:
Y e t , o n r e a d i n g i t, p e r h a p s w i t h o u t a t o n c e f u ll y u n d e r s t a n d i n g i ts
i m p o r t a n c e , m y i m p r e s s i o n w a s a b o v e a l l o n e o f s u r p ri s e.
I h a d n o i d e a t h a t s u c h s i m p l e p r i n c i p l e s , w h i c h h a d b e e n f a m i l i a r t o m e
s i n c e 1 90 7, w e r e n e w ( I a m s p e a k i n g o f t h e f ir s t t w o c h a p t e r s ; c h a p t e r 3 w h e r e
c o n t i n u e d f r a c t i o n s w e r e d i s c u s s e d w a s n e w t o m e ) .
I w a s s u r p r i s e d t h a t a s c h o l a r w h o s e w o r k o n d i v e r g e n t s e r i e s , e n t i r e
f u n c t io n s , P i c a r d ' s t h e o r e m , a n d t h e t h e o r y o f m e a s u r e I a d m i r e d , h a d g i v e n
s u c h c o m p l i c a t e d p r o o f s o f s u c h s i m p l e t h e o r e m s ( th i s t i m e I a m s p e a k i n g o f t h e
t h r e e c h a p t e r s ) . ( L e t t e r f r o m P . L E V Y , d a t e d D e c e m b e r 2 2, 1 9 69 .)
9. Early R e-workings o f Borel's Stro ng Law9.1. Introduction
T h i s c h a p t e r i s d e v o t e d t o o t h e r c o n t r ib u t i o n s f r o m t h e p e r i o d i m m e d i a t e ly
f o l lo w i n g B O R E L 's l a n d m a r k p a p e r o f 19 09 . T h e m a t h e m a t i c i a n s o f t h is t im e , w h o
w e r e a t t r a c t e d b y B O R E L ' s r e s u l t s , h e l p e d t o i l l u m i n a t e t h e r e l a t i o n b e t w e e n
m e a s u r e t h e o r y a n d p r o b a b i l i t y b y t h e i r e ff o rt s, t h o u g h t h e i r i n t e r e s t w a s p r i m a r i l y
i n t h e d i r e c t i o n o f t h e f o r m e r a n d n o t i n th e d i r e c t i o n o f p r o b a b i l i t y . T h r e e m e n
- -F A B E R , H A U S D O R FF a n d R A D E M A C H ER - - r e p r o v e d t h e B O R E L S t r o n g L a w
w i t h o u t a n y r e f e r e n c e t o t h e C e n t r a l L i m i t T h e o r e m ; i n d ee d , t h e ir m e r it ,
p a r a d o x i c a l l y , w a s t h a t t h e y d i d no t c o n c e r n t h e m s e l v e s w i t h t h e p r o b a b i t i s t i ci n t e r p r e t a t i o n o f t h e t h e o r e m . B y e x a m i n i n g t h e i r o r i g i n a l w o r k s , o n e c a n o b s e r v e
t h e s h i ft in v i e w p o i n t f r o m t h a t o f BO R EL (1 9 09 ) t o a v i e w p o i n t i n w h i c h p r o b a b i l i t y
( s p e c if i c a ll y g e o m e t r i c p r o b a b i l i t y o n [ 0 , 1 ] ) meant m e a s u r e . 1
F u r t h e r , H A U S D O RF F a l s o p r o v e d a r e s u l t o n c o n t i n u e d f r a c t io n s c l o s el y a k i n
t o t h e B E R N S T E I N - B O R E L r e s u l t , t h i s a g a i n b y i n t e r p r e t i n g t h e t h e o r e m i n t h e
s e t ti n g o f m e a s u r e t h e o r y .
T h e f ir st in o r d e r o f o c c u r r e n c e , FABER, r e g a r d e d t h e c o e x t e n s i o n o f BOREL's
d e n u m e r a b l e p r o b a b i l i ty a n d L EB ES GU E m e a s u r e a s a n o p e n q u e s t i o n w h i c h h e
w a s a t p a i n s t o r a i s e ; H A U S D O R F F w a s m u c h m o r e a s s e r t i v e a n d w e n t s o f a r a s t o
o f fe r e x p li c it l y a n " a r b i t r a r y " d e f i n i t io n o f p r o b a b i l i t y a s ( L EB E SG U E ) m e a s u r e .R A D E M A C H E R r e g a r d e d t h e d u a l v i e w p o i n t s a s s u f f i c ie n t l y w e l l a c c e p t e d a s to b e
a l m o s t w i t h o u t n e e d o f c o m m e n t . T h e w o r k o f F A BE R a n d R A D EM A C H ER (t h e
A c o m p a r i s o n b e t w e e n t h e t w o p a p e r s , F A B E R ' S d a t e d 1 9 1 0 a n d
RADEMACHER'S 1 9 1 8 , s h o w s t h e e x t e n t t o w h i c h t h e e q u i v a l e n c e b e t w e e n
L E B E S G U E m e a s u r e o n [ 0 , 1 ] a n d g e o m e t r i c p r o b a b i l i t y o n [ 0 , 1 ] h a d s h i f t e d i t s
s t a tu s f r o m c o n j e c t u r a l to a v i rt u a l ly u n a n i m o u s c o n v e n t i o n .
W h e n F AB ER , w h o s e m a i n a i m w a s in a s o m e w h a t d i f f er e n t d i r ec t io n , a c h i e v e d
a l m o s t i n a d v e r t e n t l y t h e re s u l t t h a t a l m o s t a ll n u m b e r s a r e n o r m a l ( th e " S t r o n g
L a w o f L a r g e N u m b e r s " ) , h e p a u s e d t o i n te r p o l a te s o m e c o m m e n t s ; t h e se s e rv e as a
p r i m a r y s o u r c e f o r e x a m i n i n g t h e e x t e n t t o w h i c h B O R E L ' s p a p e r l e ft i ts r e a d e r s
u n s u r e a s t o th e r e l a t io n b e t w e e n p r o b a b i l i t y t h e o r y a n d m e a s u r e t h e o r y 1:
T h e s e t o f p o i n t s f o r w h i c h l i m 7 "4 = 1 o r l i m ~ " = ~ 1, i s o f m e a s u r e z e r o .n~oo # . . . . # ,
T h i s t h e o r e m a p p e a r s i n t e re s t in g t o m e f r o m m a n y p o i n t s o f v ie w .
F i r s t i t g i v e s a s i m p l e e x a m p l e o f a s e t whi ch i s no t on l y eve rywhe re dense bu ta l so has t he card i na l i t y o f t he con t i nuum i n eve ry i n t e rva l , however sm a l l , and
none t he l e s s has m easure z e ro .
B o r e l r e c e n t l y p r o v e d , a f t e r f o r m u l a t i n g s u i t a b l e d e f in i t io n s c o n c e r n i n g
d e n u m e r a b l e p r o b a b i l it i e s , t h a t t h e p r o b a b i l i t y t h a t a p o i n t b e l o n g t o th e a b o v e
s e t is ze r o. T h e c o m p a r i s o n o f t h e a b o v e t h e o r e m w i t h B o r e l 's r e s u lt s u g g e st s t h e
q u e s t i o n :
I s t h e p r o b a b i l i t y - a c c o r d i n g t o t h e B o r e l s e t -u p w h i c h p o s s i b l y m i g h t n e e d
t o b e e x te n d e d t o a n s w e r th i s q u e s ti o n - t h a t a n u m b e r b e l o n g s t o a p r e s c r i b e d
s e t o f z e r o m e a s u r e , a l w a y s e q u a l t o z e r o ? A n d c o n v e r s e l y : is a se t a l w a y s o f
m e a s u r e z e r o , if t h e p r o b a b i l i t y t h a t a p o i n t b e l o n g s t o i t is e q u a l t o z e r o ? ( I ta l ic s
i n t he o r i g ina l . ) FABER (1910 : 400 ) .
T h e p a s s a g e j u s t c i t e d is e v i d e n c e t h a t , f o r F AB E R, t h e c l e a r - c u t i d e n t i f i c a ti o n o f
g e o m e t r i c p r o b a b i l i t y o n [ 0, 1 ] w i t h L EB ES GU E m e a s u r e h a d n o t q u i t e t a k e n p l a c e
b y 1 91 0. B O R E L 's o w n s u g g e s t i o n o f 19 05 t h a t s u c h g e o m e t r i c p r o b a b i l i t y b e
i d e n t if i e d w i t h L EB ES GU E m e a s u r e s e e m s t o h a v e b e e n u n k n o w n t o F A BE R. (R e c a l l
t h a t i n 1 9 09 t h i s s u g g e s ti o n , r e d u c e d t o a m e r e a s i d e , w a s d i s m i s s e d b y B O RE L
h i m s e l f . ) I t i s c l e a r t h a t F A B ER , o n t h e h e e l ' s o f B OR E L 'S p a p e r o f 1 90 9, w a s
g r a p p l i n g w i t h th e s a m e n a s c e n t i d e n t if i ca t io n , m o t i v a t e d b y th e " a c c i d e n t a l "
a l t e r n a t iv e a p p r o a c h h e f o u n d t o B O R E L 's S t r o n g L a w .A s q u o t a t i o n s f r o m B O RE L h a v e s h o w n , a l l t h a t w a s n e e d e d t o a n s w e r F A BE R 'S
q u e s t i o n a f f i r m a t i v e l y , a n d t o m a k e t h i s i d e n t i f i c a t i o n a b s o l u t e l y e x p l ic i t, w a s a
c a r e fu l r e - e x a m i n a t i o n o f B O R E L 's o w n p a p e r w i t h e m p h a s i s o n t h e d u a l i t y
b e t w e e n t h e " p o i n t d e v u e l o g i q u e " a n d t h e " p o i n t d e r u e g 6 o m 6 t r i q u e " .
Neit her FABER n o r BOREL c a n b e r e a l is t i c a ll y c h a r g e d w i t h o b t u s e n e s s ; BOREL
a c h i e v e d h is r e s u lt a s a n a p p l i c a t i o n o f a n a b s t r a c t t h e o r e m o n i n d e p e n d e n t t ri al s,
F AB ER b y c o n s i d e r i n g f u n c t i o n s o f a r e a l v a r i a b l e d e f i n e d o n [ 0 , 1] . T h e s i t u a t i o n
m a y b e d e s c r i b e d b y s a y i n g t h a t B OR EL , p e r f e r ri n g t h e " p o i n t d e v u e l o g iq u e ," w a s
e x p l o r i n g t h e p r o d u c t m e a s u r e a v a i l a b l e o n a c e r t a i n p r o d u c t s p a c e , a n d F A B E R
w a s e x p l o r i n g t h e c o n s i d e r a b l y m o r e f a m i l i ar t e rr i t o r y o f m e a s u r a b l e f u n c t io n sd e f i n e d o n [ 0 , 1 ] .
1 I n FABER'Snotation 7,(/~) is the num ber of l's (0%) n the first n terms of he binary exp ansion of hegiven real number.
T h e q u e s t i o n o f t h e m a p p i n g b e t w e e n t h e p r o d u c t s p a c e i m p l i c it in B OR EL a n d
t h e f a m i l i a r m e a s u r e s t r u c t u r e o f [0 , 1 ] w a s n o d o u b t o f l e s se r i n t e r e s t t o BO R EL a n d
F A B E R t h a n t h e i r p r i m a r y b u t d i s t i n c t a i m s . A l t h o u g h i t i s d i f f i c u l t f o r a
c o n t e m p o r a r y r e a d e r t o r e a d B O R EL 'S p a p e r o f 19 09 w i t h o u t e m p l o y i n g t h e
a d v a n t a g e s s u p p l i e d b y h i n d s i g h t , w e t a k e F A B E R ' s a b o v e r e m a r k a s f u r t h e r
e v i d e n c e c o n f i r m i n g o u r a s s e r t i o n t h a t B O RE L fa i le d t o s ee , a n d c e r t a i n l y t o s t a t e,
t h e i d e n t i fi c a t i o n i n 1 90 9. I n d e e d , w e t h i n k W I N T N E R w a s b e i n g s o m e w h a t o v e r -
g e n e r o u s i n h is a s s e s s m e n t o f B O RE L w h e n h e w r o t e :
H i s t o r ic a l ly , t h e w h o l e d e v e l o p m e n t w a s i n i ti a te d b y B o r e l' s f o r m u l a t i o n
a n d p r o o f o f h i s " e i t h e r 0 o r 1 " t h e o r e m ( R e n d . P a l e r m o , v o l . 2 9 (1 90 9), p p . 2 4 7
2 71 ). T o d a y i t i s e a s y , b u t a t t h a t t i m e it w a s a t r u e m a t h e m a t i c a l a c h i e v e m e n t ,
t o t h i n k o f t h e o r d i n a r y m e a s u r e o n t h e i n t e r v a l 0 < x < 1 a s a p r o d u c t m e a s u r e
i n a n i n f i n i t e p r o d u c t s p a c e , t h e f a c t o r s b e i n g t h e b i n a r y s p a c e s c o r r e s p o n d i n gt o t h e d y a d i c e x p a n s i o n o f x . W I N T N E R (1 9 4 1 : 1 8 2).
9 .3 . Radem acher and Ha usdo r f f The Evidence for t he Evo lu tion
o f a Po in t o f V iew
E i g h t y e a r s a f t e r t h e p u b l i c a t i o n o f F A B ER (1 91 0), R A D E M A CH E R , in i g n o r a n c e
o f F A B E R ' s r e s u lt , p r o v e d i t a n e w b y a v i r t u a l l y i d e n t ic a l c o m s t r u c t i o n o f a
m o n o t o n e f u n c t i o n w h o s e n o n - d i f f e r e n t i a b i l i t y w a s a s s u r e d a t t h e n o n - n o r m a l
n u m b e r s . I n t h e i n t e r im , H A U S D O R F F 'S Grundzi ige der M enge nlehre h a d a p p e a r ed .
T h i s v o l u m e h e lp e d d i s s e m i n a t e m a n y c o n c e p t s w h i c h w e r e b e c o m i n g p a r t o f t h em a t h e m a t i c a l " c u l t u r e " o f t h e t im e , f o r e x a m p l e s u c h c o n c e p t s a s fi el d s o f s e ts ,
B O R EL s e t s , l im i n f a n d l i ra s u p o f s e ts .
I n p a r t i c u l a r , H A U S D O R F F g a v e a b r i e f i n t r o d u c t i o n t o t h e L E B ES G UE t h e o r y o f
m e a s u r e a n d o f t h e L E B ES GU E i n t eg r a l . A s examples o f m e a s u r e t h e o r y H A U S -
D O R F F p r e s e n t e d b o t h t h e B O R EL r e su l t o n d e c i m a l s a n d a c l o se l y r e l a t e d
c o n t i n u e d f r a c t i o n r e s u l t , b u t w i t h p r o o f s i n n o w a y r e s t i n g o n B O R E L ' S
i n v e s t i g a t i o n o f 1 90 9 o f t h e p r o b a b i l i t y o f i n f in i te l y m a n y s u c c e s se s i n a
d e n u m e r a b l e s e q u e n c e o f t r ia l s o r i ts e x t e n s i o n i n 1 9 12 .
I n c o n t r a s t w i th t h e o r i g i n a l t r e a t m e n t o f B OR EL , b o t h t h e S t r o n g L a w a n d t h e
r e s u l t o n c o n t i n u e d f r a c t i o n s a r e t r e a t e d s o l e ly as e x a m p l e s o f t h e t h e o r y o fL E BE S GU E m e a s u r e . HAUSDORFF ' s t r e a t m e n t o f t h e s e t h e o r e m s w i ll b e g i v e n in t h e
n e x t s e c t i o n . W h a t i s n e e d e d h e r e , a s t h e n e c e s s a r y h i s t o r i c a l s e t t i n g o f
R A D E M A CH E R 'S p a p e r 1 9 18 , i s t h e p r e c i s e l a n g u a g e o f H A U S D O R F F o n t h e r e l a t i o n
b e t w e e n m e a s u r e t h e o r y a n d p r o b a b i l i s t i c t e r m i n o l o g y , a n d h i s r e s t a t e m e n t o f
B O R E L 's S t r o n g L a w . 1
W e r e m a r k t h a t m a n y t h e o r e m s c o n c e rn i n g th e m e a s u r e o f p o i n t- s et s
a p p e a r p e r h a p s m o r e i n tu i ti v el y , if o n e e x p r es s es t h e m i n t h e l a n g u a g e o f
p r o b a b i l i t y . I f t w o s e t s P a n d M a r e m e a s u r a b l e , a n d M i n p a r t i c u l a r i s o f
p o s i t iv e m e a s u r e , t h e n o n e c a n d e fi ne , b y m e a n s o f t h e q u o t i e n t f ( P ) / f ( M ) i f P
___ M , o r m or e ge ne ra l l y by f ( P c~ M ) / f ( M ) , t h e p r o b a b i l i t y t h a t a p o i n t o f Mb e l o n g s t o P . I f w e c o n s i d e r o n l y s u b s e t s o f a fi x ed s e t M , o f m e a s u r e 1, t h e n f ( P )
W e h av e taken the libe rty of altering HAUSDORFF'S original notation for u ni on and in-tersection, ~(P 1, P2) and ~(P 1, P2) respectively.
= p i s t h e p r o b a b i l i t y t h a t a p o i n t b e l o n g s t o t h e s e t P . I f P = / ' 1 w P 2, P ' = / ° 1 c~ P 2,
t h e n p + p ' = p 1 + P 2 ; P i s t h e p r o b a b i l i t y t h a t a p o i n t b e l o n g s t o P 1 o r P2, P' i s th e
p r o b a b i l i t y t h a t a p o i n t b e l o n g s t o P1 a s w e l l as t o P 2. I f P1 a n d P2 h a v e n o p o i n t s
i n c o m m o n , t h e n p = p ~ + p 2 . F u r t h e r m o r e p ' - - p 1 ~ - ( if p i > 0 ) ; t h e p r o b a b i l i -
t y t h a t a p o i n t b e l o n g s s i m u l t a n e o u s l y t o P t a n d P2 i s t h e p r o b a b i l i t y t h a t i t
b e l o n g s t o P I m u l t i p l i e d b y t h e p r o b a b i l i t y t h a t a p o i n t o f Pa s h o u l d b e l o n g t o
P 2- T h e f o r m u l a f o r s o - c a ll e d " i n d e p e n d e n t " e v e n t s p' =p ~ P2 is o f c o u r s e n o t
v a l i d i n g e n e r a l. I t i s a l s o p e r f e c t l y c l e a r t h a t f r o m t h is ( b y a n d l a r g e a r b i t r a r y )
d e f i n it i o n , p r o b a b i l i t y 0 is n o t t h e e x p r e s s i o n o f i m p o s s i b i l i t y , a n d p r o b a b i l i t y 1
n o t t h a t o f c e r t a i n t y ; f o r 0 is t h e p r o b a b i l i t y t h a t a p o i n t b e l o n g s t o a s e t o f z e r o
m e a s u r e ( w h i c h m i g h t s ti ll b e o f t h e c a r d i n a l i t y o f t h e c o n t i n u u m ) . H A U S D O R FF
( 1 9 1 4 : 4 1 6 - 4 1 7 )
T h i s c i t a t i o n s h o u l d l e a v e n o d o u b t a s to h o w e a r l y FA B ER 'S q u e s t i o n o f 1 91 0
w a s a n s w e r e d , a n d h o w e a r l y a s c r u p u l o u s l y c l e ar e x p r e ss i o n o f p r o b a b i l i t y w a s
p u b l i s h e d - i n c lu d i n g i n d e p e n d e n c e , c o u n t a b l e a d d i ti v i ty , c o n d i t i o n a l p r o b a b i l i ty ,
a n d e v e n t h e n o t i o n o f s a m p l e s p a c e . H A U S D O R F F'S r e m a r k s , p r e f a t o r y f o r h i s
e x p l i c i t r e - w o r k i n g o f B O R E L'S r e s u l t s , a r e i n t h e s a m e v e i n a s B O R EL 'S p a p e r o f
1 9 0 5 , b u t t h e y r e a c h c o n s i d e r a b l y f u r t h e r .
T w o p a g e s l a t e r , H A U S D O R F F s t a t e s BOREL's S t r o n g L a w a s f o l l o w s :
I I . Se t s o f Dyad ic Dec imals . W e c o n s i d e r a n i r r a t i o n a l n u m b e r x b e t w e e n 0
a n d 1 a n d e x p a n d i t i n a d y a d i c d e c i m a l [ f o o t n o t e o m i t t e d ] :
x 1 x2 x 3x = ~ - + ~ + 3 X + . . . . (Xx , Xz , X3 . . . . ) ( x , = 0 , 1 ) .
A m o n g t h e f i r st n d i g it s w il l a p p e a r p z e r o s a n d q = n - p o n e s. T h e n o n e h a s t h e
T h e o r e m ( E. B o r e l) :
T h e s e t o f x f o r w h i c h l i m P = l_ h a s m e a s u r e 1.
4
q 2
O r : t h e c o m p l e m e n t , i .e ., t h e s e t o f t h o s e x , f o r w h i c h p- i s e i t h e r n o n -n
c o n v e r g e n t , o r d o e s n o t c o n v e r g e t o ½, h a s m e a s u r e 0 . T h e r e i s t h u s ap r o b a b i l i t y 1 t h a t t h e d y a d i c e x p a n s i o n o f x h a s a s y m p t o t i c a l l y as m a n y
z e r o s a s o n e s .
T h i s t h e o r e m is r e m a r k a b l e . O n t h e o n e h a n d i t s e e m s a p l a u s i b l e e x te n s i o n
o f t h e " L a w o f L a r g e N u m b e r s " t o t h e i n fi n it e ; o n t he o t h e r i t as s e rt s th e
e x i s t e n c e o f a l im i t o f a s e q u e n c e , a n d i n d e e d e v e n a p r e s c r i b e d v a l u e o f t h e l i m i t,
a v e r y sp e c i al c i rc u m s t a n c e , w h i c h o n e w o u l d h a v e h e l d a pr ior i t o b e
e x c e e d i n g l y u n l i k e l y . H A U S D O R F F ( 1 9 1 4 : 4 1 9 - 4 2 0 ) .
W e m a y t h u s s t a te t h a t f o r H A U SD O R F F t h e n o t i o n o f p r o b a b i l i t y is
e m p h a t i c a l l y s e e n a s a d e r i v e d o n e , r e s t i n g o n t h e m o r e f u n d a m e n t a l o n e o f
m e a s u r e . T h e s u b s e q u e n t p r o o f d o e s n o t r e f e r t o p r o b a b i l it i e s f o r n o t a t i o n ,
i n t u i ti o n o r m e t h o d , b u t e m p l o y e d s o le l y t h e l e n g t h s o f i n t e rv a l s o f c o v e r i n g s e ts
( se e b e lo w ) . T h e t h e o r e m s e r v e d H A U SD O R F F o n l y a s a n e x a m p l e , a l t h o u g h a v e r y
i n t e r e s t in g o n e , o f th e u s e o f c o u n t a b l e a d d i t i v i t y o f L EB ES G UE m e a s u r e .
His p roo fs are a l l in the langu age of (LEBESGUE) mea sure , w i th m od er n
not at ion s , such as l im sup E, for a sequ ence {En} of se ts . As to the r e la t io n o f his
(measure - theore t i c ) a s se r t ions to those of BOREL, which were couched in the
language of probabi l i ty , HAUSDORFF expl i c i tly d isposes of th is in the pre f a toryr e ma ks c i te d a bove .
10. Hausdorff's "Grundziige der Mengenlehre":
A Notable Advance in Technique
10.1. General Background
This sect ion wi l l c i te vi r tual ly a l l the probabi l is t ic references to be found in
HAUSDORFF's Grundzfige der Mengenlehre. The f i r s t p robabi l i s t i c re fe rence ,
a l r e a dy c i te d , c onc e r ns t he r e l a t i on be t w e e n t he voc a bu l a r y o f p r oba b i l i t y a nd t ha t
of me asur e the ory. In this sect ion we discuss HAUSDORFF's t re atm en t of dy adic
e xpa ns i ons a nd c on t i nue d f r a c t i ons v i a me a s u r e t he o r y .
C ha p t e r X o f t he Grundziige (1914) is the locu s o f a l l of the abo ve i tems. T his
chap te r i s t i t led "C on ten t o f Poin t -Se t s ." I ts f i rs t pa r agr aph begins wi th genera l i t ie s
concerning length, measure and area in EUCLIDEAN space, and a his tor ical sketch.
Th e olde r the or y of con ten t asso cia ted wi th CANTOR, HANKEL, PEANO, and
JORDAN, conc ern ed i t se l f wi th f ini te add i t ivi ty fo r dis joint se ts ; the newe r theo ry,
due to BOREL and LEBESGUE, specif ical ly ad ded co un tab le ad di t ivi ty. HAUS-
DORFF ci ted LEBESGUE'S pr ob lem and for m ula ted i t thus :
. .. L e be s gue f o r m u l a t e d t he p r ob l e m o f a s s oci a ti ng t o e ve r y bou nde d s et A in n -d i me ns i ona l s pa c e E , , a s " c on t e n t , " a num be r f (A)> 0 sat is fying the fol lowing
c ond i t i ons :
(a ) Congruent s e t s have the same measure .
( f i ) The uni t cube has conten t 1 .
(7) f( A + B) = f( A) + f(B).(6) f (A + B + C +...) = f(A) + f(B ) + f( C) +... f o r a boun de d s um o f c oun -
tably many sets . HAUSDORFF (1914: 401).
HAUSDORFF imm edia te ly gave an exam ple show ing the im poss ib i l i ty of th i s
problem in fu l l genera l i ty . The example , s t i l l t he mos t popula r one (cf ROYDEN
(1965: 52-55) , map s the l ine on to th e c i rc le ; on the c i rc le i t exhibi ts a se t which is
d i s jo in t f rom a l l i t s images under ra t iona l ro ta t ions . Fur the r , th i s s e t and i t s
(necessa r i ly congruent ) images under a l l ra t iona l ro ta t ions i s a d i s jo in t decom-
pos i t ion of the en t i re c i rc le .
I t is of interes t to n ote th at this ex am ple is no t a t t r ibu ted by HAUSDORFF to a ny
a u t ho r . H ow e ve r , i n t he r e f e r enc e s ne a r t he e nd o f t he boo k t w o s ou r c es o f exa mpl e s
of non -m easu rabl e se ts a re g iven: G . VITALI 's Sul ProbIema della Misura dei Gruppi
di Punti di una Retta, a nd A . SCHOENFLIES'Mengenlehre. T he l a t t e r c on t a i n s
severa l examples of non -me asur able set s, inc luding the o ne g iven by HAUSDORFF.
Th e text of SCHOENFLIES (SCHOENFLIES (1913: 377, especia l ly fo otn ote 2))
indicates interes t ingly that this s imple example was in fact due to HAUSDORFF
himse l f , and communica ted d i rec t ly .
HAUSDORFF fur the r com me nte d on the im poss ib i l i ty o f the " easy LEBESGUE
problem" (employing the t e rminology of NATANSON (1961: 80) ) in which
L e t A ~ ( e ) = P(E(e)) = m e a s u r e o f E (~ ). T h e s e ts ~ ) E , ( e) d e c r e a s e w i t h i n c r e a s i n gn = N
N . O b v i o u s l y
E(e) c ~) E,(e) for N = 1 , 2 , 3 , . . . .n ~ N
F o r e a c h f i x e d i n t e g e r N w e h a v e , b y m o n o t o n i c i t y a n d c o u n t a b l e s u b - a d d i t i v i t y ,
t h a t
P(E,(e)) = p,(~).~(e)<=p E,(e) <=
H a v i n g sh o w n th a t ~ pn(e ) co n v er g es fo r ev e r y e > 0 , t h e m e as u re E (e ) = A oo(e) i s 0
f o r e v e r y e > 0 ) , =N
I n p a r t i c u l a r ~ = ~ ) E ( ~ ) i s a c o u n t a b l e u n i o n o f s e ts o f m e a s u r e 0 , a n d b y
co u n ta b le ad d i t iv i ty , i s i t s e l f o f m e as u r e 0 . B u t l=~)1E (~) i s t h e se t o f x fo r w h ich
l i m n 2 > 0 . T h u s i f ~ p , ( e ) c o n v e r g e s f o r a l l e > 0 , th e r e i s p r o b a b i l i t y (i.e.,n~oo
m e a s u r e ) 1 t h a t :
= 0limbo n 2
o r e q u i v a l e n t l y
v,(x) _1_J i m n 2 = 0 .
A t t h e v e r y l e a s t , H A U S D O R F F g a v e t h e f i r s t (c f p r e c e d i n g f o o t n o t e )
" C A N T E L L I " - t y p e p r o o f t h a t ~ p , < o o i m p l i e s A o o = 0 . I t s e e m s c l e a r t h a t
H A U S D O R F F , n o t C A NT E LL I, s h o u l d b e g i v e n c r e d i t f o r t h i s l i n e o f r e a s o n i n g . N o t
o n l y d i d H A U S D O R F F a n t i c i p a t e C A N T E LL I b y t h r e e y e a r s , h e a l s o g i v e s a t r e a t m e n t
1 This same argument shows (suppressing the e), that if ~p ,< o o , where p,=P(E,), thenP(lim sup E,) = A ~ = 0 thanks to sub additivity and with no appea l to independence. In other words, this
is CANTELLI'S argument, but resting on the firm base of measure theory. CANTELLIhad rested thecorresponding argument on the ad hoe assumption that sub-additivity("BooLn'S inequality") could be
extended from the finite to the countably infinite case.As early as 1906, in his thesis, FRECHET prov ed a theorem to which th e "CANTELLI" esult is an
imm ediate corollary, but couched in the language of measure (cf FRgCHET 1906: 16)). If for each integern, E, is a measurable subset of the unit interval whose measure re(El)equals 1 -m l , then FR£CHET
°bserved that m(~]lE't>l-(m~+m2+'")"~ / n It is immediate (although it seems to have
/
passed
unrem arked in the literature) that i f ~ m, converges (the only case of interest) then m E, > 1 - e forn=
N sufficiently large, depending on e. This yields m(lim inf E,) = 1 at once w ithout the intervention of any"independence" assumption. Since m,=m(E~,), his can be rephrased as follows: ~m(U,) convergesimplies re(lira inf U, )= 0, precisely th e "CANTELLI"result.
H A U S D OR F F s h o w e d g e n e r a ll y h o w t o e x p re s s " m o m e n t s " s u c h a s
k = o ( 2 k - - n ) l 2
a s p o l y n o m i a l s i n n w i t h i n t e g e r c o e f f i c i e n t s . T h e e v e n v a l u e s o f l a r e t h e m o s t
s i g n if i ca n t , as in t h e c a s e l = 4 c it e d a b o v e . T h e s e p o l y n o m i a l s a r e e a s i l y b o u n d e d
a b o v e b y p u r e p o w e r s o f n w i t h a s l ig h t ly l a r g e r l e a d i n g c o e ff ic i en t . T h u s
H A U S D O R F F s h o w s h o w t o o b t a i n e s t i m a t e s s u c h a s
a n d h e n c e
A2mk=O
¢ 1k~0
k l >I f 2 ., m e a n s s u m o v e r t h o s e v a l u e s o f k s a t i s f y i n g = e t h e r e r e s u l t s f i n a l ly
2
e 2 , , ~ 1 < A 2 ~ 12 n n m 22m •
I n s h o w i ng h o w t he s e m o m e n t s c o u l d b e e s ti m a t e d a n d h o w a d v a n t a g e o u s t h ey
w e r e, H A U SD O R F F a c h ie v e d a n o t a b l e a d v a n c e i n m e t h o d . T h e a b i li ty t o m a k e s u c he s t im a t e s w a s d e s t i n e d t o p l a y a r o le i n l a t e r d e v e l o p m e n t s b o t h w h e n l = 4 a n d
w h e n l = 2 m f o r l a r g e m . I n d e e d , C A N T E L L I w a s t o m a k e u s e , t h r e e y e a r s l a t e r , o f
s i m i l a r t e c h n i q u e s i n v o l v i n g t h e 4 t h m o m e n t , i . e . , l = 4 , a n d S TE IN H A US c o n s i d -
e r e d l = 2 m f o r l a r g e v a l u e s o f m i n 1 92 3.
T o c o n c l u d e t h e s e r e m a r k s , w e s k e tc h H A U S D O R FF 'S m e t h o d o f o b t a i n i n g e x a c t
p o l y n o m i a l e x p r e s s i o n s f o r
HAUSDORFF f i r s t c o n s i d e r e d
k=0
a n d i n t r o d u c e d t h e c h a n g e o f v a r i a b l e s u = x + y , v = x - y . T h e n t h e d i f f e r e n t i a l
0o p e ra to r D = x ~xx - y ffyy sa t i s f ies
D ( f + g ) = D f + D g ,
D ( f g ) = ( O f ) g + f ( D g ),
D ( f ' ) = n f ~ - 1 (D f ) ,
D ( x k y . - k) = (2 k - n ) x k y" - k .
I n p a r t i c u l a r , D ( u ) = v , D ( v ) = u .
T h e c a l c u l a t io n o f D2m(un) t h e n p r e s e n t s n o d i f fi cu l ty , a n d t h e n u m e r i c a l c h o i c e
x = y = ½ , w h i c h c o r r e s p o n d s t o u = 1 , v = 0 , r e s u l t s i n
i (2k-nl. . . . . 1 = \ k ] 2 " 'v = 0 k = O
an m th d e g r e e p o l y n o m i a l i n n.
1 0 . 4 . H a u s d o r f f ' s C o n t i n u e d F r a c t i o n T h e o r e m
L e t u s i n t r o d u c e o n c e m o r e t h e n o t a t i o n [ a 1 = m l , a 2 = m 2 . . . . . a , = m , ] t o s t a n d
f o r t h e s e t o f c o n t i n u e d f r a c t i o n s w h o s e f ir s t n e l e m e n t s a l , . . . , a , a r e t h e p r e s c r i b e d
i n t e g e r s m l . . . . , m , . T h e s e f o r m a n i n t e rv a l , t h e l e n g t h o f w h i c h w e d e n o t e b y
P [ a l = m l , . . . , a n = m , ] .
T h e s e t d e f i n e d b y
[ a l = m l , . . . , a , _ l = m , _ l , k < = a , < m ]
i s a l so a n i n t e r v a l ( b e i n g a u n i o n o f a d j a c e n t i n t e r v a ls o f t h e a b o v e t y p e ) , th e l e n g t h
is d e n o t e d b y
P [ a 1 = m 1 . . . , a , _ 1 = m , _ 1 , k < a , < m ] .
T h e r a t i o o f t h e a b o v e t o t h e l e n g t h P [ a l = m l , . . . , a , _ 1 = l ~ n - - 1] c a n b e d e n o t e d b y
P [ k < a , < m [ a l = m l , . . . , a ~ _ l = m ~ _ l ]
u s i n g t h e m o d e r n n o t a t i o n f o r c o n d i t i o n a l p r o b a b i l i t y a l t h o u g h t h i s n o t a t i o n i s n o ts t r i c t ly n e c e s s a r y ( a n d i n d e e d is n o t u s e d b y HAUSDORFF). H A U S D O R F F 'S p o i n t o f
d e p a r t u r e is a n i n e q u a l i t y
p ( k , m ) < P [ k < a , < m l a I = m I . . . . , a , _ 1 = m , _ 1] < ~ r ( k , m )
w h e r e p a n d a c a n b e c a l c u l a t e d e x p l ic i tl y , a n d d o n o t d e p e n d o n n , n o r o n
m l , . . . , m , _ 1 . H A U S D O R F F c a l c u l a t e s " b e s t p o s s i b l e " e x p r e s s i o n s f o r p a n d a ,
i n c l u d i n g t h e i r s o m e w h a t a l t e r e d f o r m i f k = 1 o r m = o e ( b u t n o t b o t h ) . I t f o l l o w s
r e a d i l y t h a t
p ( k , , m , ,) < P [ k , < = a , , < m , , [ k 1 < =a 1 < = m I . . . . , k n - 1 < = a n - l < = m , - 1] ~ O'(kn, rn,)
b y p u r e l y a l g e b ra i c m a n i p u l a t i o n s . W e h a v e d e s c r i b e d t h i s m a n i p u l a t i o n i n
d i s c u s s i n g B E R N S T E I N 's w o r k ; i t i s e s s e n t i a l l y i d e n t i c a l i n H A U S D O R F F 'S .
B y m u l t i p l i c a t i o n o f t h e s e i n e q u a l i t i e s f o r c o n s e c u t i v e v a l u e s o f n ( w h i c h c a n b e
i n t e r p r e t e d a s t h e C h a i n L a w o f P r o b a b i l i t y ) i t f o l lo w s t h a t
P l . .. P , < P [ k l < a l < m a , . . . , k n < a , < m , ] < 1 1 . . . ~r,
w h e r e w e h a v e i n t r o d u c e d
p j = p ( k ~ , m j ) , a j = a ( k ~ , m j )
f o r b r e v i t y .
B y c o u n t a b l e a d d i t i v i t y ( e x p l ic i t l y s o s ta t e d )
co
] ~ p j < = P [ k j < =aj < = m j , j = l , 2 , 3 . . . . ] ~ ~ I a j .1 1
H A U S D O R F F is i n t e r e s te d o n l y in c o n d i t i o n s o n t h e t w o n u m e r i c a l s e q u e n c e s { k ,}
a n d { m , } t h a t a s s u r e p o s i t i v e m e a s u r e t o t h e s e t [ k j < a j < m j , j = 1 , 2 , 3 , . . . ] . F o r t h i soo oo
i t i s c l e a r l y n e c e s s a r y t h a t I ~ a j > 0 a n d s u f f ic i e n t t h a t H p j > 0 . I n v i e w o f t h e s p e c i f i c1 1
f o r m o f p ( k , m ) , a ( k , m ) i t r e s u l t s t h a t p o s i t i v e m e a s u r e i s o b t a i n e d i f a n d o n l y i f k j = 1
e x c e p t f o r f i n i t e l y m a n y v a l u e s o f j a n d
1 c o n v e r g e s .j = l m r
H A U S D O R F F d i d n o t p u r s u e h i s c a l c u l a t i o n s f u r th e r , a s h a d B E R N S T E IN a n d
B O R E L , to o b s e r v e t h a t t h e s e t d e f i n e d b y [ k n < a n < m n f o r a ll b u t f i n it e l y m a n y
v a l u e s o f n = 1, 2 , 3 , . . . ] i s n e c e s s a r i l y o f m e a s u r e 1 w h e n i t i s p o s i t i v e ( a l t h o u g h t h e
r e s u l t is i m m e d i a t e u s i n g h i s t e c h n i q u e s ) . R e c a l l , i t w a s th i s r e s u l t w h i c h s oa p p e a l e d t o B O R E L t h a t h e h a d p o i n t e d i t o u t a s th e m o s t i n t e r e s t i n g i n h is e n t i re
p a p e r o f 1 90 9.
I n s u m m a r y , H A U S D O R F F u s e d c o n t i n u e d f r a c t i on s a s a n e x a m p l e o f t h e p o w e r
o f L E B ES G U E m e a s u r e , e s p e c i a l l y i ts c o u n t a b l e a d d i t i v i ty a n d w i d e d o m a i n o f
d e f i n i t io n . H e o b t a i n e d a r e s u l t a l o n g t h e l i n e s o f t h e B E R N S T E I N -B O R E L o n e . L i k e
B E R N S T EIN h e d id n o t m i s t a k e n l y a s s u m e o r e m p l o y i n d e p e n d e n c e , b u t r a t h e r
m a n i p u l a t e d t h e p r o b a b i l i t ie s o f m u t u a l l y d e p e n d e n t e v e n t s in a f u ll y a c c u r a t e
m a n n e r . L i k e B E R N ST E IN , H A U S D O R F F o b t a i n e d t h e p r o b a b i l i t i e s o f n o n - c y l i n d e r
s et s b y v a l id l im i t o p e r a t i o n s o n t h e p r o b a b i l i t y o f a p p r o x i m a t i n g c y l i n d e r s et s.
N o n e o f B O R E L ' s e a r l y s u c c e s s o r s f o l l o w e d h i s l e a d t o w a r d s t h e r i g o r o u sf u n d a t i o n o f a t h e o r y o f p r o b a b i l i t y c o n c e r n e d w i t h r e p e a t e d t r ia l s ( i n d e p e n d e n t o r
n o t) . N o t u n t i l s o m e o n e w h o p o s s e s s e d f ac i li ty w i t h a x i o m a t i c a l l y b a s e d m e a s u r e
t h e o r y a n d w h o s h a r e d h i s p r i m a r y c o n c e r n w i th p r o b a b i l i t y ( in p a r t i c u l a r w i t h
r e p e a t e d t r i a ls ) w a s B O R E L t o h a v e a s u c c e s s o r a s i m p o r t a n t a s h i m s e lf , a s u c c e s s o r
w h o w o u l d f u l ly d e s e r v e t h e a c c o l a d e w h i c h W I N T N E R b e s t o w e d o n B O R EL . T h a t
w a s t o w a i t u n t i l 1 9 23 ; t h e s u c c e s s o r w a s t o b e H U G O S T E IN H A U S .
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