Các dạng hội tụ của dãy hàm đo được

8/2/2019 Cc dng hi t ca dy hm o c

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Hunh Vit Khnh SP. Ton 01-K.30 - 1 -


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LI NI UCc dng hi t ca dy hm o c l mt phn

nh trong lnh vc o v tch phn Lebesgue. y l

mt trong cc mng gii tch c ng dng nhiu trongthc t, v c bit l nn tng cho gii tch hin i. Do

, vic nghin cu v n l rt cn thit.

V thi gian hon thnh lun vn ny tng i

ngn nn khng th nghin cu su hn, v chc cn

nhiu sai st. Rt mong nhn c s gp ca qu thy

c v qu bn c.

Em xin chn thnh cm n B mn Ton to

iu kin cho em nghin cu. Xin cm n c Trn Th

Thanh Thy nhit tnh hng dn v gip em sa cha

kp thi cc sai st trong lun vn ny.

Sinh vin thc hin

Hunh Vit Khnh


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NHN XT CA GIO VIN HNG DN

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Cn Th, ngy thngnm 2008

Trn Th Thanh Thy


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NHN XT CA GIO VIN PHN BIN

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Cn Th, ngy.. thng.. nm 2008


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MC LC

MU................................................................................................................. 71. L do chn ti.............................................................................................. 72. Gii hn ca ti............................................................................................ 7

3. Mc tiu ti.................................................................................................. 7

NI DUNG ............................................................................................................. 9

Chng 1: KIN THC CHUN B........................................................................ 91.1 O.......................................................................................................... 9

1.1.1 i s tp hp.......................................................................................... 91.1.2. - i s ................................................................................................. 91.1.3. - i s Borel...................................................................................... 101.1.4. o trn mt i s tp hp ............................................................... 111.1.5 Mrng o ....................................................................................... 13

1.1.6 o trn r ......................................................................................... 151.2- HM SO C .................................................................................. 171.2.1 nh ngha ............................................................................................. 171.2.2 Mt s tnh cht ca hm so c ..................................................... 181.2.3 Cc php ton trn cc hm so c.................................................. 20

1.3- TCH PHN LEBESGUE .......................................................................... 231.3.1. Tch phn ca hm n gin khng m ................................................. 231.3.2 Tch phn ca hm o c khng m................................................... 241.3.3 Tch phn ca hm o c bt k ......................................................... 261.3.4 Tnh cht................................................................................................ 26

1.3.5 Gii hn qua du tch phn..................................................................... 27Chng 2: SHI TCA DY HMOC ............................................. 30

2.1 CC DNG HI T CA DY HM O C.................................... 302.1.1Hi t hu khp ni (converges almost everywhere) .............................. 302.1.2Hi tu (converges uniformly) ........................................................... 312.1.3Hi tu hu khp ni (converges uniformly almost everywhere)........ 322.1.4Hi t theo o (converges in measure) .............................................. 322.1.5Hi t trung bnh (converges in the mean) ............................................. 342.1.6Hi t hu nhu (converges almost uniformly) .................................. 35

2.2 CC DNG DY C BN ........................................................................ 36

2.2.1Dy cbn hu khp ni (Cauchy almost everywhere, hoc fundamentalalmost everywhere)......................................................................................... 362.2.2 Dy cbn u ( uniformly Cauchy)..................................................... 372.2.3 Dy cbn hu nhu (almost uniformly Cauchy)............................. 372.2.4 Dy hm cbn trung bnh (Cauchy in the mean hoc mean fundamental)....................................................................................................................... 372.2.5Dy cbn trong o (Cauchy in measure, hoc fundamental inmeasure)......................................................................................................... 37


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2.3 SLIN H GIA CC DNG HI T CA DY HM O C...382.3.1 Lin h gia hi t trung bnh v hi t theo o ................................ 382.3.2 Lin h gia hi t trung bnh v hi t hu khp ni ........................... 392.3.3 Lin h gia hi t theo o v hi t hu khp ni ............................ 402.3.4 Lin h gia hi t trung bnh v hi tu........................................... 43

2.3.5 Lin h gia hi t hu nhu v hi t hu khp ni ......................... 432.3.6 Lin h gia hi t theo o v hi t hu nhu ............................. 452.3.8 Lin h gia hi t hu khp ni v hi tu....................................... 482.3.9 Lin h gia hi t trung bnh v cbn trung bnh............................... 492.3.10 Lin h gia cbn trung bnh v cbn theo o........................... 502.3.11 Lin h gia cbn trung bnh v hi t hu nhu.......................... 502.3.12 Lin h gia cbn hu nhu v hi t hu nhu....................... 502.3.13 Lin h gia cbn theo o v cbn hu nhu........................ 522.3.14 Lin h gia cbn theo o v hi t theo o ............................ 532.3.15 Lc th hin mi lin h gia cc dng hi t................................ 54

Chng 4: BI TP............................................................................................... 56KT LUN........................................................................................................... 72

TI LIU THAM KHO........................................................................................ 73


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LUN VN TT NGHIP


MU

1. L do chn ti

o v tch phn Lebesgue l nn tng ca gii tch hin i. Vic nghin

cu n l cn thit, gip cho em nm vng hn kin thc v phn ny. Ngoi ra, em

cn c iu kin nghin cu su hn cc mng gii tch c lin quan. y l l do

chnh em chn ti ny.

2. Gii hn ca ti

o v tch phn Lebesgue l mng gii tch hin i kh rng. Trong

khung kh mt lun vn tt nghip, ti khng th khai thc mi vn . Do vy,

lun vn tp trung khai thc v mt s dng hi t ca dy hm o c. Bn cnh

, cn xt v mi lin h gia cc dng hi t ny.

3. Mc tiu ti

Trong phm vi gii hn ca ti, mc tiu hng ti ca lun vn l nghin

cu mt s dng hi t ca dy hm o c. C th hn, bn cnh cc dng hi t

quen thuc nhhi t theo o, hi t hu khp ni, ti cn nghin cu mt s

dng hi t khc nh hi t hu nhu, hi tu hu khp ni, hi t trung

bnh,

Tuy nhin, hiu su hn v cc dng hi t, ti cn tp trung nghin

cu v mi lin h gia cc dng hi t ny. V d, nh ta bit, trong khng gian

o hu hn v o c xt l o th mi dy hm o c hi t hu

khp ni th hi t theo o. Vn t ra l i vi cc dng hi t khc th c

mi lin h vi nhau nh th no? V cc mi lin h ny c thay i hay khng khi


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LUN VN TT NGHIP


ta xt chng trong khng gian o hu hn? ti s tp trung lm r cc vn

ny.

thun tin trong qu trnh nghin cu, lun vn cn cp n mt s

khi nim mi nh dy cbn theo o, dy cbn trung bnh,V khng ngoil, lun vn cng cp n mi lin h gia cc khi nim ny.


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NI DUNG

Chng 1: KIN THC CHUN B.

1.1 O

1.1.1 i s tp hp

nh ngha

Mt i s (hay trng) l mt lp nhng tp cha X, v kn i vi miphp ton hu hn v tp hp (php hp, php giao hu hn cc tp hp, php hiu

v hiu i xng hai tp hp).

nh l 1

Mt lp tp hp l mt i skhi v ch khi C tha mn cc iu kin sau:

a. C ;

b. A C CA C ;

c. BA, C BA C .

1.1.2. - i s

nh ngha

Mt - i s (hay - trng) l mt lp tp hp cha ,A v kn i vi mi

php ton m c hay hu hn v tp hp.

nh l 2

Mt lp tp hp F l mt -i skhi v ch khi F tha mn cc iu kin sau:


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LUN VN TT NGHIP


a. F ;

b. A F CA F ;

c. nA F 1

nn

A

=

U F .

Nhn xt

Mt - i shin nhin l mt i s.

nh l 3

Cho M l mt h khng rng cc tp con ca X.

a. Lun tn ti duy nht mt i s ( )C M bao hm M v cha trong tt c

cc i skhc bao hm M i s ( )C M gi l i ssinh bi M .

b. Lun tn ti duy nht mt - i s ( )F M bao hm M v cha trong tt

c cc - i s khc bao hm M - i s ( )F M c gi l - i s

sinh bi M .

1.1.3. - i s Borel

nh ngha

Cho khng gian tp ,(X ) . - i s sinh bi h tt c cc tp m trong X

c gi l - i sborel.

K hiu: ( )XB .

Nhn xt

Cc tp m, tp ng l cc tp Borel.

Nu , 1,2,...nA n = l cc tp Borel th1

nn

A

=U v

1n

n

A

=I , theo th t l


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LUN VN TT NGHIP


cc tp kiu F ,G cng l nhng tp Borel.

- i sBorel trong mt khng gian tp Xcng l - i snh

nht bao hm lp cc tp ng.

1.1.4. o trn mt i s tp hp

nh ngha

Cho C l mt i strn X.

Hm tp hp : C R l mt o trn C nu:

a. ( ) 0A , A C .

b. ( ) 0; =

c. =

=U

1nnA ( )

1n

n

A

= , vi ( ) , , .n m nA A m n A n = C

o c gi l hu hn nu ( ) +


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LUN VN TT NGHIP


Cc tnh cht ca o

nh l 4

Cho l o trn i sC .a. BA, C , ( ) ( )B A B A ;

b. BA, C , ( ) +


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LUN VN TT NGHIP


nh l 5

Cho l o trn i sC .

a. Nu ( ) ,0=iA 1 ii A

= U C th .01 =

=Ui iA

b. NuAC , ( ) 0=B th ( ) ( ) ( )\BA B A A = = .

nh l 6

Cho l o trn i sC .

a. Nu nA C ( ),n ....,21 AA1

n

n

A

=

U

C th ( )1

lim .n nnn

A A

=

=

U

b. Nu nA C ( ),n ....,21 AA ( ) ,1 +


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LUN VN TT NGHIP


a. ( )* 0, ;A A X

b. ( )* 0; =

c.1

n

n

A A

=

U ( ) ( )* *

1

n

n

A A

=

(tnh cht - bn cng tnh).

nh l 8 (nh l Carathodory)

Cho * l mt o ngoi trn ,X L l lp cc tp con A caXsao cho:

( ) ( ) ( )* * * \ ,E E A E A E X = + ( )1

Khi :

a. L l mt - i s.b. * = | L l mt o trn L .

o ny c gi l o cm sinh bi o ngoi * .

Tp A tha ( )1 c gi l * -o c.

nh l 9

Cho m l mt o trn mt i sC

nhng tp con ca X.Nu vi mi A X t:

( )* infA = ( )1 1

, ,i i ii i

m P P A P

= =

U C ( )2

Khi :

a. * l o ngoi;

b. * | L m= ;

c. C ( )F C .L

nh l 10

Nu l cm sinh bi o ngoi * th:


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LUN VN TT NGHIP


a. H cc tp c o bng 0 trng vi h tp c o ngoi *

bng 0.

b. l o .

nh l 11 (mrng o)

Cho m l o trn mt i s L Khi , tn ti mt o trn - i s

L ( )F C C , sao cho:

a. ( ) ( );A m A =

b. l hu hn (- hu hn) nu m l hu hn (- hu hn);

c. l o ;d. A L khi v ch khiA biu din c di dng:

\A B N = hoc A B N =

Trong B ( )F C , N E ( )F C , ( ) ( )* 0E E = = v * l o ngoi xc

nh t m bi cng thc ( )2 .

Nhn xt

L sai khc ( )F C mt b phn cc tp c o khng, tc l - i s L cc tpo c c th thu c t ( )F C bng cch thm hay bt mt b phn ca mt tp

c o khng.

1.1.6 o trn r

Ta gi gian trn ng thng l mt tp hp c mt trong cc dng

sau:[ ] ( )( ] [ ) ( ) ( ) ( ) ( ] [ ), , , , , , , , , , , , , , , ,a b a b a b a b a a a a + + + .

Xy dng i s

Gi C l lp tt c cc tp con ca c th biu din thnh hp ca mt s hu

hn cc gian i mt ri nhau, tc l:


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LUN VN TT NGHIP


( )1

: , ,n

i i ii

P P i j n=

= =

UC r= r r r .

Khi , C l mt i s.

Nu P C v ( )1

, ,

n

i i ji

P i j=

= = Ur r r t ( ) 1

n

ii

m P== r .

Khi , m l o trn C v m l o - hu hn.

Mrng o

Vi ,A R o ngoi c xc nh bi:

( ) ( )*

1 1

inf , ,i i ii i

A m P P A P

= =

=

U C

iu ny c th thay bng:

( )*1 1

inf , ,k k kk i

A A

= =

=

Ur r r

Gi L l tp tt c cc tp con A ca sao cho:

( ) ( ) ( )* * * ,E E A E A E = + \ .

o mrng trn - i s L c gi l o Lebesgue.

Cc tp A L c gi l nhng tp o c theo ngha Lebesgue (hayA o

c ( )L .

Nhn xt

o Lebesgue trn l - hu hn v [ ]1

,n

n n

=

U= v hin nhin

. l o . Mi tp Borel trn u o c Lebesgue.

Tp o c Lebesgue chnh l tp Borel thm hay bt mt tp c .

. o khng.


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LUN VN TT NGHIP


1.2- HM SO C

1.2.1 nh ngha

Cho tp ,X F l mt i snhng tp con ca X, v AF .

Khng gian ( ),X F dc gi l khng gian o c.

Mt hm s :f A c gi l o c trn tp A i vi i s F nu:

( ){ }, : .a x A f x a < F

Hay vit gn l:

{ },A

a f a < FR .

Nu trn F c o th f c gi l o c i vi o hay o

c.

Nu ,k=F L v kX = th ta ni f o c theo ngha Lebesgue hay ( )L - o

c.

Nu k=F B , v kX = th ta ni f o c theo ngha Borel hay f l hm s

Borel.

Nhn xt.

Hm s f o c trn A ( )+ ,1 af F , a .

nh l 1

Cho ( ),X F l khng gian o c v hm :f X . Khi , cc iu kin sau

l tng ng.

( )i Hm f o c trn A

( )ii { },A

a f a < F

( )iii { }, Aa f a F

( )iv { },A

a f a > F


18/73

LUN VN TT NGHIP


( )v { },A

a f a F

Chng minh

( ) ( )i ii : Do nh ngha.

( ) ( )ii iii : ( ) ( ) 1, ,a f x a f x a nn

< +

{ }1

1.

An A

f a f an

=

= < +

I F

( ) ( )iii iv : ,a t:

{ }A

M f a=

{ }AN f a= <

Ta c:

M N A = , v M N =

N A M = \

N F .

( ) ( )iv v : Ta c:

,a ( ) ( )1

:f x a n f x a

n

>

{ }1

1.

An A

f a f an

=

=

I F

( ) ( )v i : ,a t:

{ }A

D f a=

{ }A

E f a= <

E A D= \ A D= F

Vy fo c.

1.2.2 Mt s tnh cht ca hm so c


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LUN VN TT NGHIP


( )i Gi s f o c trn A . Nu B A , B F th f cng o c trnB .

Tht vy, v B A , v B F nn: ,a

{ } { } .B A

f a B f a< = < F

Vy, f o c trn .B

( )ii Nu f o c trn A th { }, .A

a f a = F

Tht vy,

,a R { } { } { }A A A

f a f a f a= =

Do : { }A

f a= F .

( )iii Nu ( ) ,f x c x A= th fo c trn .A

Tht vy, ,a ta thy:

{ }, ;

, .Ac a

f aA c a

< =

F

Nu 0k> th { } .A

akf a f

k < = <

F

Do , vi 0k ta c kf o c trn A .

Nu 0k= th ( )( ) 0, .kf x x A=

Do , theo ( )iii , ta c kf o c trn A .

Nh vy, hm kfo c trn A .

( )v Nu f o c trn { }n nA (hu hn hoc m c) th f o c trn


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LUN VN TT NGHIP


j nn

AU .

Tht vy, ,a ta c:

{ } { }n nn A Anf a f a

< = < U UF

Vy, f o c trn .nn

AU

( )vi Nu f xc nh trn A , ( ) 0A = v th f o c trn A .

Tht vy, ,a ta c:

{ }A

f a A<

Do ( ) 0,A = v nn { }Af a< F Vy, f o c trn A .

1.2.3 Cc php ton trn cc hm so c

Cho ( ),X F l khng gian o c, A F .

( )i Nu f o c trn A th vi 0 > , hm f

o c trn A .

Tht vy, vi 0 > , ta c:

{ } { } { } , a 0;

, 0.AA A

f af a f a a

< = > < >

{ }A

f a < F

Vy,

f o c trn .A

Tuy nhin, mnh o ca mnh trn ni chung khng ng. Ngha .l, c th xy ra trng hp f

nhng f khng o c.

V d: Xt hm s:

( )1, ;

1, .

x Af x

x A

=


21/73

LUN VN TT NGHIP


Trong ,A , A l mt tp khng o c Lebesgue.

Ta c 11

, .2

f A + =

Do , fkhng o c trn .

Nhng, ( ) 1,f x x= R nn f o c trn .

( )ii Nu f v g o c, hu hn trn A th f g+ o c trn A .

Gi { }n nr l dy cc s hu t.

,a f g a+ < f a g <

: .nn f r a g < <

{ } { } { }( )1 .n nA A Anf g a f r g a r + < < < U F

f g + o c trn .A

Tuy nhin, mnh o ca mnh trn ni chung khng ng. Ngha

l, nu ta c f g+ o c th cha suy ra c f v g o c.

V d: Xt cc hm s

( )

=Ax

Axxf

,0

,1v ( )

1, ;

0, .

x Ag x

x A

=

Vi ,A A l tp khng o c Lebesgue.

Ta c:

( )1 ,0f A =

v ( ) Ag =+ ,01 .

nn gf, l nhng hm s khng o c trn .

Nhng, ( )( ) 0,f g x x+ = R nn gf + o c trn .

( )iii Nu f v g o c v hu hn trn A th f g cng o c trn A .

Tht vy, v g o c nn g o c. Do , ( )f g f g = + o

c trn A .

( )iv Nu f v g o c, hu hn trn A th .f g o c trn A .


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LUN VN TT NGHIP


Tht vy, ( ) ( )2 21

.4

f g f g f g = + nn .f g o c trn .A

Tuy nhin, mnh o ca mnh ( )iv khng ng.

V d: Xt cc hm s

( )1, ;

0, .

x Af x

x A

=

v ( )0, ;

1, .

x Ag x

x A

=

Vi ,A A l tp khng o c Lebesgue.

R rng, ,f g khng o c trn .

Nhng, ( )( ). 0,f g x x= nn gf. l hm o c trn .

Nhn xt:

Hm f o cA hmf+ v f o c trn .A

Trong :

{ }max ,0 ;f f+ = { }max ,0f f =

( )v Nu f v g o c, hu hn trn A th { } { }max , , min ,f g f g o c

trn .A

Tht vy, ta c:

{ } ( )1max , 2f g f g f g = + + ;

{ } ( )1

min ,2

f g f g f g = + l nhng hm o c trn .A

Do { }min ,f g , { }max ,f g o c trn A .

( )vi Nu f v g o c v hu hn trn A , ( ) 0, ,g x x A thf

go

. c trn .A

Tht vy, do ( ) 0,g x x A nn:

,a 22

, 0;1

1, 0

AA

a

ag ag

a

< = > >


23/73

LUN VN TT NGHIP


2

1.

A

ag

<

F

Nh vy,2

1

go c trn .A

Do2

1.

ff g

g g= nn suy ra

g

fo c trn .A

( )vii Nu dy ( ){ }n nf x N l mt dy nhng hm so c v hu hn trn

A th cc hm s ( ){ }sup n nn

f xN

, ( ){ }inf n nn f x N , ( ){ }lim n nf x N ,

( ){ }lim n nf x N l nhng hm o c, v nu tn ti lim nx f f = , th f

cng o c trn A .Tht vy,

,a R ( ){ }{ } { }1

sup n n An A n

f x a f a

= I F

,a R ( ){ }{ } { }1

inf n n An A nf x a f a

< = < U F

Do , ( ){ }sup n nn

f xN

, ( ){ }inf n nn f x N l nhng hm o c trn .A

V 1lim inf sup ;n mn m nf f = 1lim sup inf n mm nnf f=

Nn suy ra nn ff lim,lim cng l nhng hm o c trn .A

Do , nu ffnn

=

lim th lim nf f=

Vy, f o c trn A .

1.3- TCH PHN LEBESGUE

1.3.1. Tch phn ca hm n gin khng m

nh ngha

Xt mt khng gian c o ( ), ,X F , AF .


24/73

LUN VN TT NGHIP


Hm s f xc nh trn A c gi l hm n gin nu fo c v nhn

mt s hu hn nhng gi tr hu hn.

Nh vy, nu f l hm n gin khng m xc nh trn tp .A F Khi , f

c dng:

( )=

=n

iA xaf i

1

( )*

Trong , iA o c, ri nhau v Un

iiAA

1=

= .

Ngi ta gi ( )=

n

iii Aa

1

l tch phn ca hm n gin fi vi o trn .A

K hiu: Afd

.Tch phn ca hm n gin khng m f c xc nh bi ( )* l duy nht vi

mi cch biu din ca hm f .

1.3.2 Tch phn ca hm o c khng m

Trc khi trnh by nh ngha tch phn hm o c khng m, lun vn

cp li nh l v cu trc ca hm o c:

nh l

Mi mt hm so c trn A u l gii hn ca mt dy { }n nf nhng hm

n gin trn A : lim , .nn

f f x A

=

Hn na, nu 0,f th tn ti { }n nf sao cho:

nf n gin, 0nf , 1n nf f+ , v lim , .nn f f a =

Chng minh

Ta chng minh cho trng hp 0f trn .A

Vi mi s t nhin n , ta t:


25/73

LUN VN TT NGHIP


( ){ }nxfAxCn = :0

( ) ( )1

: , 1, 2,..., 22 2

i nn n n

i iC x A f x i

= < =

t:

( )

0,

1,

2

n

n inn

n x Cf x i

x C

=

Khi , nf l hm n gin trn A , 0nf , v 1n nf f+

Ta chng minh lim nn

f f

=

+ Nu ( )


26/73

LUN VN TT NGHIP


Gi s f l o c khng m, xc nh trn tp A . Khi , tn ti nf

dy hm n gin, khng m, n iu tng, v lim .nx

f f

=

Tch phn ca hm f trn A i vi o c nh ngha l:

fd = lim nnA

f d .

1.3.3 Tch phn ca hm o c bt k

Nu f l hm o c bt k, ta phn tch: + = fff .

NuA

f d+ hocA

f d hu hn th hiu sA

f d+ A

f d c ngha v n

c gi l tch phn ca hm f trn A i vi o .

Hm f c gi l kh tch trn A nu A

fd hu hn.

1.3.4 Tnh cht

( )i Cc tnh cht n gin:

( ).A cd c A = ( ) ( ).B

A

x d A B =

( ) ( )1 1

.i

n n

i B i ii iA

x d A B = =

=

( )ii Nu f o c trn A v ( ) 0A = th 0.A

fd =

( )iii Nu f o c, gii ni trn A v ( )A < th fkh tch trn A .( )iv Tnh cht cng tnh:

Nu A B = thA B A B

fd fd fd

= + , nu mt trong hai v ca

ng thc c ngha.


27/73

LUN VN TT NGHIP


( )v Tnh bo ton th t:

Nu gf = h.k.n trn A th .A A

fd gd =

( gf = h.k.n trn A nu ( ) 0: = BAB v ( ) ( ) B\, Axxgxf = ).

Nu f g trn A thA A

fd gd

( )vi Tnh cht tuyn tnh:

,A A

cfd c gd c = .

( ) .A A A

f g d fd gd + = + (nu v phi c ngha).

1.3.5 Gii hn qua du tch phn

nh l hi tn iu

Cho dy hm o c { }nf .

Nu 0 nf f trn A th lim nn

A A

f d fd

= .

Chng minh Nu { }nf l dy cc hm n gin th hin nhin theo nh ngha tch phn ta

c lim .nn

A A

f d fd

=

Xt trng hp { }nf bt k.

Gi ijh l cc hm n gin khng m sao cho :

11131211 ... fhhhh n

22232221 ... fhhhh n

.

t { }1 2max , ,...,n n n nnh h h h=

Ta c: nh l dy hm n gin, khng m, n iu tng v nn fh


28/73

LUN VN TT NGHIP


Do , ( )*n nA A A

h f f

Mc khc, nk th nnkn fhh

Cho n ta c : fhf nn

k

lim

Cho k , ta c fhnn

=

lim

Kt hp vi (*), v cho n ta suy ra : lim nn

A A

f d fd

= .

B Fatou

Nu 0nf trn A th lim lim .n nA

f d f d

Chng minh

t { },...,inf 1+= nnn ffg

Ta c nn fg lim0

Do , =A

n

A

nn

fg limlim

Nhng v nn fg , n nn A

n

A

n fg

Do : = A

n

A

nn

A

n fgg limlimlim

Vy, lim lim .n nA

f d f d

Ch :

( )i Nu ggfn , kh tch trn A th b Fatou vn cn ng.

Tht vy, do gfn nn 0 gfn trn A.

T kt qu trn ta c:

( ) ( ) A

n

A

n gfgf limlim

V A

g hu hn nn :


29/73

LUN VN TT NGHIP


( ) ( ) ++AA

n

A A

n ggfggf limlim

hay, A

n

A

n ff limlim .

( )ii Nu ggfn , kh tch trn A th:

A

n

A

n ff limlim .

Tht vy,

Do gfn nn gfn v do hm g kh tch nn tho cu a. ta c:

( ) ( ) A

n

A

n ff limlim

( ) A nA n ff limlim Vy,

A

n

A

n ff limlim


30/73

LUN VN TT NGHIP


Chng 2: S HI T CA DY HM O

C

2.1 CC DNG HI T CA DY HM O C

2.1.1Hi t hu khp ni (converges almost everywhere)

nh ngha

Cho dy { }n nf v hm f o c trn A .

Dy { }n nf c gi l hi t hu khp ni v hm f trnA nu:

:B A ( ) 0B = v ( ) ( )lim nn

f x f x

= , \Bx A

K hiu: .a enf f hay. . .h k nnf f

V d:Xt cc hm s:

1 2,

, 3nn n

f n

= v 0f = trn [ ]0,1

Khi .a enf f trn [ ]0,1 .

Tnh cht

Cho { } gff nn ,, l nhng hm o c trn A . Khi , nu ffea

n . trn

A v .a enf g trn A th gf = h.k.n trn .A

Chng minh

V ff ean . trn A nn ( ) ( ) ( ): 0, lim , \Bn

nB A B f x f x x A

= =

V .a enf g trn A nn ( ) ( ) ( ): 0, lim , \Cnn

C A C f x g x x A

= =

Gi CBD =

Ta c: ( ) 0=D , v D\Ax th ( ) ( )xgxf =

Vy, gf = h.k.n trn .A


31/73

LUN VN TT NGHIP


Nh vy, nu ng nht cc hm s tng ng th gii hn h.k.n ca mt

dy nhng hm so c l duy nht.

nh l hi t b chn ca Lebesgue

Nu , ,nf g n hm g kh tch, v.a e

nf f (hoc trong o) trn A th:lim .nn

A A

f f

=

2.1.2Hi tu (converges uniformly)

nh ngha


Dy { }n nf c gi l hi tu v hm f trn A nu:

( ) ( ) ( )0 0 00, : , nn n n n x A f x f x > = <

K hiu: nf fI .

V d

Trn ( )( ), , P vi l o m.

Xt dy hm ( ) 1 , 1

0 ,n

x nf x xx n

= >

Ta c ( )nf x I1

xtrn .

Tht vy, 0 01

0, , ,n n n x

> > > ta lun c:

( )0

1 1nf x

x n

< .

Vi { } ,n nf f l nhng hm o c trn A .

t ( ) { }n kk n

T r f f r

=

= >U

Khi ( )nT r l tp o c.


32/73

LUN VN TT NGHIP


Sau y l iu kin cn v mt dy hm o c hi tu.

nh l

Cho { } ,n nf f l nhng hm o c trn tp .A

Khi nf fI ( ) ( ) ( )0, : n rr n r T r > = .

Chng minh

( ) Gi s nf fI trn A

Khi :

( ) ( )0, : ,r n r k n r x A > th ( ) ( )kf x f x r

( ) ( )n rT r = .

( ) Gi s ta c ( )0, :r n r > ( ) ( )n rT r =

( ) ,k n r x A th ( ) ( )kf x f x r

Vy nf fI trnA .

2.1.3 Hi t u hu khp ni (converges uniformly almost

gfhfdheverywhere)


Dy { }n nf c gi l hi tu hu khp ni v hm f trnA nu:

( ): 0B A B = v nf fI trn \B.A

K hiu: nf fI hu khp ni (hoc nf fI a.e).

2.1.4Hi t theo o (converges in measure)

nh ngha


Dy { }n nf c gi l hi t theo o v hm f trnA nu:


33/73

LUN VN TT NGHIP


( ) ( ){ }0, lim : 0nn

x A f x f x

> =

K hiu: nf f trn A .

Nh vy,

nf f ( ) ( ){ }0 00, 0, 0 : , : nn n n x A f x f x > > > < .

V d: Xt cc hm s:

1 2,

n

n n

f

= , vi 2n v 0f = trn [ ]0,1

Khi nf f trm [ ]0.1 , vi l o Lebesgue trn .

Tnh cht ca dy hmhi t theo o

Cho l o.( )i Nu nf f

trn A , l o , v f g= h.k.n th .nf g

( )ii Nu nf f v nf g

th f g= h.k.n.

Chng minh

( )i Ta c: n nf g f f f g +

{ } { } { }0, 0n nf g f f f g > >

{ } { } { }0 0n nf g f f f g + >

V f g= h.k.n nn: { }0 0f g > =

Do , { } { }0 0n nf g f f khi n

Vy, .nf g

( )ii Ta c: n nf g f f f g +

Do vy, v

i0,

>

2

2

n

n

f ff g

f g

Do ,


34/73

LUN VN TT NGHIP


{ }2 2n n

f g f f f g

Suy ra,

{ } 2 2n nf g f f f g

+

Khi n th:

2nf f

0

2nf g

0

Suy ra { } 0f g = khi n

c bit,1

, 0n f gn

=

Mt khc,

{ } { }1

10

n

f g f g f gn

= > =

U

Suy ra { }1

10

n

f g f gn

=

Vy, f g= h.k.n.

2.1.5Hi t trung bnh (converges in the mean)

nh ngha

Dy { }n nf cc hm kh tch c gi l hi t trung bnh v hm kh tch f

trn A nu:

( ), 0n nA

f f f f d = khi n .


35/73

LUN VN TT NGHIP


nh l hi t trung bnh (Mean convergence theorem)

Gi s { }n nf , f l nhng hm o c trn A tha mn:

( ) ( )lim ,nn

f x f x x A

= , v tn ti nhng hm kh tch ,h g tha mn:

( ) ( ) ( ) nAxxhxfxg n ,, ( )1

Khi , nf f kh tch v nf hi t trung bnh v f .

Chng minh

Do { }n nf , f l nhng hm o c nn nf f cng o c.

T ( )1 cho n ta c: ( ) ( ) ( ) ,g x f x h x x A

( ) nh g f f h g

0 nf f h g , vi h g l hm kh tch.

Kt hp vi iu kin ( ) ( )lim ,nn

f x f x x A

= , p dng nh l hi t b chn

ca Lebesgue, ta suy ra:

lim 0nn

A

f f

=

Vy, nf hi t trung bnh v f .

2.1.6Hi t hu nhu (converges almost uniformly)

nh ngha

Dy hm o c { }n nf c gi l hi t hu nhu v hm o c f

trnA nu:

0, > ( ):B A B < v nf fI trn \BA

K hiu: .a unf f .


36/73

LUN VN TT NGHIP


nh l

Cho dy { }n nf v hm f o c trn A . Khi :

( )( ) 0. rTff nua

n khi n .

Chng minh

( ) 0 > cho trc, B o c: ( )B < , v nf fI trn \A E

Do , vi 0r> , ta c:

( ) ( ) ( ) ( ) ( )( ): n r n r n r T r B T r <

Do 0 > l ty nn( ) ( )( ) 0n rT r khi n .

( ) Do ( )( ) 0nT r khi n nn:

0 > cho trc, v vi mi s t nhin p sao cho:

1:

2pp n pn T

p

= ( ) ( ) 0, ,n mf x f x n m n < .


37/73

LUN VN TT NGHIP


2.2.2 Dycbn u ( uniformly Cauchy)

Dy hm o c { }n nf trn A c gi l cbn u trn A nu:

( ) ( )0 00, : , , .m nn m n n x A f x f x > <

2.2.3 Dycbn hu nhu (almost uniformly Cauchy)

Dy hm o c { }n nf trn A c gi l cbn hu nh u trn A nu:

( )0, :E E > < v { }n nf cbn u trn \E .A

2.2.4 Dy hm cbn trung bnh (Cauchy in the mean hoc meanfundamental)

Dy { }n nf cc hm kh tch c gi l cbn trung bnh nu:

( ), 0n m n mA

f f f f d = khi ,m n .

2.2.5Dy cbn trong o (Cauchy in measure, hoc fundamental

in measure).

Cho khng gian o ( ), , ,X A F F .

Cho dy { }n nf o c trn A vi l o.

Dy { }n nf c gi l cbn theo o trnA nu:

{ }0, 0n mf f > khi , .m n


38/73

LUN VN TT NGHIP


2.3 SLIN H GIA CC DNG HI T CA DY HM

O C

2.3.1 Lin h giahi t trung bnh vhi t theo o

nh l

Cho dy hm { }n nf , v hm f kh tch trn A . Khi nu nf hi t trung

bnh v f th ffn .

Chng minh

Vi s dng ty 0 > cho trc, t:

{ }n AB f f =

Ta lun c: ( )n nA B

f f d f f d B

T gi thit nf ht t trung bnh v hm f , ta c: 0nA

f f d khi n

Do , ( ) 0B khi n

Nh vy, nf f .

Chiu ngc li ca nh l ni chung khng ng, tc l, nu dy hm

nf f , th ta cha th suy ra c nf ht t trung bnh v hm f . Sau y l

mt v d minh ha.

V d:

Xt dy hm nf trn [ ]1,0 c xc nh nh sau:

( )

1, 0,

10, ,1

n

n xn

f xx

n

=

v hm 0f = .

Ta c: [ ]{ }( )1 1

0,1 : 0; 0nx f f n n

= =

khi .n


39/73

LUN VN TT NGHIP


Do , .nf f

Tuy nhin, [ ]

11

0 0

1, 0,1n

nf f d nd x = =

nn nf khng hi t trung bnh v hm f .

2.3.2 Lin h giahi t trung bnh vhi t hu khp ni

nh l

Cho dy hm { }n nf kh tch trn A . Ngoi ra,.a e

nf f . Khi nu tn ti

hm kh tch khng m g sao cho nf g th nf hi t trung bnh v hm f .Chng minh

Do .a enf f nn. 0a enf f

Ta cn c: n nf f f f + 2g

p dng nh l hi t b chn ta c:

0nA

f f khi n

Vy, nf hi t trung bnh v hm f .

V d sau cho thy { }n nf hi t hu khp ni v hm f , nhng { }n nf

dfghdfkhng hift trung bnh v hm f .

Xt ( ) 2 21nn

f xn x

=+

.

Ta c: [ ]1,0x ,

( ) 2 22

1lim lim lim 011nn n nnf x n x nxn

= = =+ +

Do , . 0a enf trn on [ ]1,0 .


40/73

LUN VN TT NGHIP


Tuy nhin, ( )1 1

2 20 0

lim lim1nn n

nf x dx dx

n x =

+

[ ] ( )020

1lim lim lim

1 2

nu nxn

n n n

du arctgu arctgnu

=

= = = =

+

Nh vy, nf khng hi t trung bnh v 0.

nh l

Cho dy hm { }n nf , v fkh tch.

Nu nf hi t trung bnh v hm f th tn ti dy con { } { }kn nf f sao cho

.

.ka e

nf f Chng minh:

Do nf hi t trung bnh v hm f nn nf f

Suy ra tn ti dy con { } { }kn n

f f sao cho . .k

a enf f

2.3.3 Lin h giahi t theo o vhi t hu khp ni

nh l

Cho dy hm { }n nf o c trn A , hi t hu khp ni v f trn A v

l o th f o c trn A .

Nu ( )A < th .nf f

Chng minh

Ta chng minh f o c trn A .t { }: nB x A f f = / , khi ( ) 0B =

Do nn f o c trn B .

Mt khc, trn \B,A nf f

f o c trn \B.A


41/73

LUN VN TT NGHIP


Nh vy, fo c trn ( )\BA B A= .

Gi s ta c ( )A < , ta cn chng minh nf f trn .A

Vi 0 > cho trc, ta c:

Nu , : n in i f f + th x B

{ }1 1

n in i

f f B+

I U

Do nn { }1 1

0n in i

f f +

=

I U

t { }1

n n ii

E f f +

= U

Nh vy, ta c:1

0nn

E

= I

Khi n tng, s hng n if + gim. Do , s phn t ca A trong nE b t i. V

th, ta c 1 2 3 ... ...nE E E E

Mt khc, ( ) ( )1E A < . Do ta c ( )lim 0nn E =

( )1lim 0nnE

=

Ta c:

{ } { } { }1 1 11 2

n n i n n ii i

E f f f f f f + +

= =

U U

Ta c: { }( ) ( )10 n nf f E

{ }( )lim 0nn

f f

=

Vy, .nf f

C nhng dy hm hi t hu khp ni nhng khng hi t hu theo o.

V d

Xt tp hp s thc vi o Lebesgue.

Chn[ ], 1n n n

f += .


42/73

LUN VN TT NGHIP


Ta c 0nf khi n

Tuy nhin, nf khng hi t hu theo o v 0 v vi1

2 = th:

( ) [ ]1

0 , 1 1 02x f x n n

= + = .

nh l

Cho dy hm { }n nf , v hm f o c trn A . Khi , nu nf f trn

A th tn ti dy con { } .: .k k

a en nn

f f f

Chng minh

Chn dy s dng { } : 0,k kk v dy { }1

: 0, .k k kkk

=

> <

Do nf f nn ( ) ( ) { }, : ,k n k k n k n n k f f

t ( ) ( ){ }1 2 11 , max 1, 2 ,...n n n n n= = +

1 2 ...n n < < v { }( ), kn k kk f f

t { },ki n kk iQ f f

== U v

1i

iB Q

== I

( ) ( ) { }( ) 0ki n k k k i k i

B Q f f

= =

( ) 0B =

Mt khc, \Bx A : ii x Q

, 0kn k

k i f f khi k

Vy . .k

a enf f


43/73

LUN VN TT NGHIP


2.3.4 Lin h giahi t trung bnh v hi tu

nh l

Nu { }n nf l dy hm kh tch hi tu v hm f trn A , v ( )A < + ,

th nf hi t trung bnh v hm f .

Chng minh

Do nf fI nn: ( ) ( )0 00, : , ,nn f x f x n n x A > <

Ta c n nf f f f +

nn n nA A A

f f f f +

( ). A + nA

f < +

Vy, f kh tch trn A .

Mt khc, 0 ,n n ta c:

( ) , 0nA

f f A >

0, .nA

f f n

Vy, nf hi t trung bnh v hm f .

2.3.5 Lin h giahi t hu nhu vhi t hu khp ni

nh l

Cho { } ,n nf f l nhng hm o c trn A . Khi nu.a u

nf f trn A

th .a enf f trn .A

Chng minh

Do .a unf f nn vi mi s t nhin k, tn ti kE sao cho ( )1

kE k < v

nf fI trn .ckE


44/73

LUN VN TT NGHIP


t1

ck

k

F E

=

= U th nf f trn F

Mt khc, ( ) ( )1

,c kF E kk <

( ) 0cF =

Vy, .a enf f trn A .

Chiu ngc li ca nh l ni chung khng ng. Tuy nhin nu c thm

mt siu kin, th chiu ngc li sng. C th, ta c nh l sau:

nh l

Cho { } ,n nf f l nhng hm o c trn A , ( )A < + . Khi nu

.a enf f th

.a unf f .

Chng minh

Vi ,k n l nhng s nguyn dng cho trc, t:

1kn m

m n

E f fk

=

= cho trc, ( )kn: \E ,2k kkn A n n

<

t1

k

kn

k

F E

=

= I , th Fo c, v:

( ) ( ) ( )k

kn n

11

\F \E \Ek

kk

A A A

==

=


45/73

LUN VN TT NGHIP


2.3.6 Lin h giahi t theo o vhi t hu nhu

nh l

Cho dy hm { }n nf , f l nhng hm o c trn A . Khi , nu

.a unf f th nf f

.

Chng minh

Do .a unf f nn vi mi s t nhin m , tn ti tp o c mE tha:

( )1

mE m < , v nf fI trn m\EA

( ) ( ), : ,N m n N m > th { }n mf f E { }( ) ( )

1,n mf f E mm

<

{ }( ) 0nf f =

Vy, nf f .

Chiu ngc li ca nh l ny ni chung khng ng. Tuy nhin, ta c

nh l sau:

nh l

Cho dy hm { }n nf , f l nhng hm o c trn A .

Nu nf f , th tn ti dy con { }

kn kf { }n nf sao cho

.

k

a unf f .

Chng minh

Do nf f nn ta c:

k , t ( ) ( )1

n nE x f x f xk

=

th ( ) 0nE khi n

Chn kn sao cho kn n > ta c ( ) 2k

nE<


46/73

LUN VN TT NGHIP


t ( ) ( )1

kk nA x f x f x

k

=

v j k

k j

G A

=

= U

Khi , vi C Cj kx G x A , vi k j

Suy ra, ( ) ( ) 1knf x f x k < , vik j

t jG G= , ta c: ( ) ( ) ( )12 2 ,k jj k

k j k j

G G A j

+

= =

= =

Suy ra, ( ) 0G = .

Nu ,jx G x G j ( ) ( )1

knf x f x

k < , k

Vi 0 > , chn k sao cho1k < . Khi ta c:

,x G l kn n > ( ) ( )1

lnf x f x

k < <

Do kn khng ph thuc vo x nn ta c.

k

a unf f .

nh l Riesz

Nu { }n nf l dy hm o c l cbn theo o trn ( ),A A < + , th tnti dy con { }

kn kf , v hm o c f sao cho

knf hu khp ni, hu nhu, v

theo o v f trn .A

Chng minh

Do dy hm o c { }n nf l cbn trong o, do :

( ) ( ) { }( )0, : , m nN m n N f f >

t ;21

1

=Nn

1

1max 1, , 1

2k k kn n N k +

= +

Xt dyknk

fg = l dy con ca { }nnf , dy hm { }kkg c tnh cht sau:


47/73

LUN VN TT NGHIP


Nu t

= + kkkk ggE 2

11 th ( )

1

2k kE

t I U

=

=

=1n nk

kEF th ( ) 0=F

Mt khc, \ , : \ ,x k xx A F k x A E k k

Do vy, kij > , ta c:

( ) ( ) ( ) ( ) ( ) ( )xgxgxgxgxgxg iijjij ++ + 11 ...

1

1 1...

2 2j i< + +

1

1

2i<

( )xgk l dy cbn trong , \R x A F

( )xgk l dy hi t

t:

( )( )lim , \

0 , \

kk

g x x A F f x

x A F

=

Nh vy, fg eak . trn .A

Vi 0> cho trc, v s t nhin k tha mn1

1

2k .

t: U

=

=kj

kk EF

Khi : ( ) th:

( ) ( ) ( ) ( ) ( ) ( )xgxgxgxgxgxg iijjij ++ + 11 ...

ij 2

1...

2

11

++


48/73

LUN VN TT NGHIP


12

1 i

Cho j ta c: ( ) ( ) 1 11 1

, ,2 2

Ci Ki k

f x g x x F i k <

Suy ra, ig hi tu v f trn ckF

Vy, fg uai . trn .A

Mi dy hi thu nhu th hi t hu khp ni. Do , ta c th chon

dy con { }iig ta c .fgi

2.3.8 Lin h giahi t hu khp ni vhi tu

nh l Egoroff

Cho ( ) , tn ti ( ): CE A E < , v nf hi tu v hm f trn E.

Chng minh

Do .a enf f trn A nn:

C A : ( ) 0C = v ( ) ( )lim nn

f x f x

= \x B A C =

Vi mi ,m n , t:

( ) ( ),1

:m n jj n

B x B f x f xm

=

=


49/73

LUN VN TT NGHIP


t ,1

\mm n

m

D A B

=

= I

Ta c: ( ) ( ),1 1

\2mm n mm m

D B B

= =


50/73

LUN VN TT NGHIP


2.3.10 Lin h giacbn trung bnh vcbn theo o

Nu dy { }n nf cc hm kh tch l c bn trung bnh th { }n nf l c bn

trong o.

Chng minh

Vi s dng ty 0 > cho trc, t:

{ }mn n m AA f f =

Ta lun c: ( )mn

n m n m mn

A A

f f d f f d A >

Do { }n nf l dy cbn trung bnh, suy ra 0n mA

f f d khi ,n m

( ) 0mnA khi , .n m

Nh vy, { }n nf l cbn trong o.

2.3.11 Lin h giacbn trung bnh vhi t hu nhu

nh l

Cho { }n n

f l nhng hm kh tch trn A . Khi nu { }n n

f l cbn trung

bnh trn A th tn ti dy con { }kn k

f ca dy { }n nf , v hm o c f trn A sao

cho .k

a unf f trn A .

Chng minh

V dy { }n nf l cbn trung bnh trn nn { }n nf l cbn trong o.

Theo nh l Riesz, tn ti dy con { }kn k

f ca { }n nf , v hm f sao cho

.k

a unf f trn A .

2.3.12 Lin h giacbn hu nhu vhi t hu nhu

nh l


51/73

LUN VN TT NGHIP


a. Dy hm o c { }n nf tha.a u

nf f trn A th { }n nf l dy cbn hu

nhu trn .A

b. Nu { }n nf l cbn hu nhu trn A th tn ti hm o c f sao cho

.a unf f

Chng minh

a. Do .a unf f nn ( )0, :B A B > < , v nf fI trn \B.A

Khi ,

( ) ( )0 00, : , \B2nn n n f x f x x A

> > <

Nh vy, 0 00, : ,n n m n > > , ta c:

( ) ( ) ( ) ( ) ( ) ( ) , \Bm n m nf x f x f x f x f x f x x A + <

{ }n nf l cbn u trn \BA .

Vy, { }n nf l cbn hu nh u trn .A

b. Ly1

, 1, 2,...kk

= =

Khi , kE : ( )1

kEk

< v { }nn

f l cbn u trn \ kA E

t1

kk

E E

=

= I

Ta c: ( )1

,E kk

<

Do : ( ) 0E =

Nu :c Ckx E k x E

( ){ }n nf x l dy cbn

Do , tn ti ( )f x : ( ) ( )lim nn

f x f x

=

Ta xem ( ) 0f x = trn E

Ta c: .a enf f


52/73

LUN VN TT NGHIP


Vi 0, 0, ,F n > sao cho:

( )F < v ( ) ( ) 0, , ,C

n mf x f x x F m n n < >

t CG F E= ta c ( )G <

Cho m ta c ( ) ( ) 0, ,nf x f x x G n n <

Do nf fI trnCG

Nh vy .a unf f .

2.3.13 Lin h giacbn theo o vcbn hu nhu

Cho dy hm { }nnf o c trn A . Khi , nu { }nnf l cbn theo o

th tn ti dy conknk

f sao choknk

f l cbn hu nhu.

Chng minh

Do { }nnf l cbn trong o trn A nn , :kk m

1 1, ,

2 2m n kk kf f m n m

<

( )1

t: ;11 mn =

{ };,1max 212 mnn += { };,,1max 323 mnn +=

Khi , dy s .....,, 21 nn s xc inh cho ta dy con knkf ca { }nnf

Vi mi k, t:

1

1

2k kk n n kE f f

+

=

,i kk i

B E i

=

= U

Ta chng minhkn

f l cbn u trn ciB .

Vi mi i v s thc 0> bt k, ta chn sr, sao cho ,isr v 11

2s <


53/73

LUN VN TT NGHIP


Vi mi Cix B , ta c:

1r s k k n n n nk s

f f f f +

=

12kk s

=

121

s

<

Nh vy, dy conknk

f l cbn u trn ciB

Mt khc, 0> ,1

1:2i

i <

Kt hp vi ( )1 ta c:

( ) ( ) 11 1

2 2i k k ik i k iB E

= =

= cho trc, ta lun c:

m n m nf f f f f f +

{ }2 2m n m n

f f f f f f


54/73

LUN VN TT NGHIP


{ }( )2 2m n m n

f f f f f f

+

Cho ,m n , kt hp vi iu kin nf f trn A , ta c:

{ }( )m nf f 0

Vy, { }n nf l dy cbn trong o.

( ) Gi s{ }n nf l cbn trong o.

Khi , tn ti dy con l { }kn k

f sao cho { }kn k

f l dy cbn hu nh

u.

Gi ( ) ( )limknk

f x f x

= , vi x m ti gii hn tn ti.

Ta c, vi mi 0 > :

{ }2 2k kn n n n

f f f f f f

Suy ra:

{ }( ) 0nf f khi n

Vy, .nf f

2.3.15 Lc th hin mi lin h gia cc dng hi t.

Nh vy, s hi t ca cc dy hm o c c mi quan h vi nhau. d

nm c cc mi quan h ny, sau y l lc th hin mi quan h gia chng.

Tuy nhin, mi quan h ny c s thay i, ty thuc c hay khng iu kin

( )A < . Sau y l hai trng hp ny:

K hiu: uni: hi tu (uniformly).au: hi t hu nh u (almost uniformly).

a.e: hi t hu khp ni (almost everywhere).

meas: hi t trong o (in measure).

mean: hi t trung bnh (in mean).


55/73

LUN VN TT NGHIP


Trng hp ( )A <

Trng hp tng qut

a.e

mean

a.u

meas

uni

mean

a.u

meas a.e

uni


56/73

LUN VN TT NGHIP


Chng 3: BI TP

Bi gii

Ta nh ngha hm nf nh sau:

( )

, tn ti s 00

1:n x

n<

Khi : 0n n > ta c:1

xn

<

( ) 0nf x = < .

Tuy nhin, ta c:

[ ] 011,0

1,0 /== nn nddf khi n

Vy, { }nnf khng hi t trung bnh v .f

Bi gii

Trn khng gian o [ ]1,0=X vi o Lebesgue trn r , t:

( ) nn xxf = v ( ) 0=xf .

( )1 Chng minh ffn hu nhdu.

Bi 1: Cho v d v dy hm { } [ ]1,01Lf nn sao cho.a e

nf f nhng { }nnf

khng hi t trung bnh v .f

Bi 2: Cho v d v dy hm o c tha mn cc iu kin sau:

( )1 Hi thu nhu.

( )2 Khng hi tu.


57/73

LUN VN TT NGHIP


Vi 0> , t{ }

2

1,min = , v [ ]1,1 =A

Ta c:

( )m A = < , ( ) ( ) ( )n

n xfxf 1 , \Ax X

Vy, nf hi t hu nhu trn X.

( )2 Chng minh nf khng hi tu v f trn X .

Ta thy ti 1=x th ( ) 1xfn nn ffn /

Vy, nf khng hi tu v hm f trn [ ]0,1 .

Bi gii

( ) V nf f nn:

( ) ( ){ }0, 0, , : : nN n N x f x f x > > <

Chn = ta c:

( ) ( ){ }0, , : : nN n N x f x f x > <

( ) 0, 0, :mm

> > <

Vi 0m

> theo gi thit ta c:

, :N n N ( ) ( ): nx f x f xm m

<


58/73

LUN VN TT NGHIP


Bi giiTa c: nf f

trn ( ) ( ){ }0, lim : 0nn

x f x f x

> =

( ) ( ){ }0, , : : nN n N x f x f x > =

( ) ( )0, , , : nN n N x f x f x >


59/73

LUN VN TT NGHIP


T ( )1 v ( )2 ta c:

=AA

nn

ff .lim

Bi gii

( )a V lin tc nn:

( ) ( ){ } ( ) ( ){ }: :n nx f x f x x f x f x / /o o

( ) ( ){ }( ) ( ) ( ){ }( ): :n nx f x f x x f x f x / /o o

T gi thit: ff ean . suy ra:

( ) ( ){ }( ): nx f x f x / 0=

( ) ( ){ }( ): nx f x f x /o o 0=

Vy, . .a enf f o o

( )b Do l lin tc u nn:

( ) ( ) ( ) ( ) .:0,0 yxyx

Trng hp ffn u.

Do nf hi tu v hm f nn:

( ) ( ) ( ) ( ) > xfxfAxnnn n,:,0 00

Do , ( )( ) ( )( ) 0, ,nf x f x n n x A < > o o .

Vy, nf f o o u trn A .

Trng hp ffn hu nhu.

Bi 6: Cho dy hm { }nn

f , v hm f nhng hm o c trn A, v hm

: . Chng minh rng:

( )a Nu lin tc v ff ean . th . .a enf f o o

( )b Nu lin tc u v ffn u, hu nhu, hay theo o th

ffn .. u, hu u, hay theo o.


60/73

LUN VN TT NGHIP


Do ffn hu nh u trn A nn: Vi 01 > , E : ( ) 1 > < <

Do nu:

( ) ( ) ( ) ( )n nf x f x f x f x < cho trc, t:

{ }= ffE nn

Khi :

Bi 7: Cho ( ),X l khng gian o tha mn ( )


61/73

LUN VN TT NGHIP


( ) +

++

=

nn EX n

n

E n

nn ff

ff

ff

ffffd

\ 11,

( ) ( )\n nE X E +

Do l s dng ty , v ( ) nn 0 > , ta c:

( ) ffd n , +

nE n

n

ff

ff

1( )

1 nE

+

V ( ) 0, ffd n nn ( )nE 0

Vy, nf f

Bi gii

Cch 1:

V dy hi tu v hm f nn N nguyn dng tha:

1,nf f n N <

Nh vy, 1+< Nff

Hm 1+Nf l hm kh tch v Nf l hm kh tch v ( ) ( )

0 0: ,nn f f x X n nX

< >

Mt khc, 0nn , ta c :

Bi 8: Gi s ( ),, FX l mt khng gian o hu hn . Gi thit { }nf l dy

hm kh tch hi tu v hm f . Chng minh rng:

=XX

nn

fddf lim


62/73

LUN VN TT NGHIP


( )( )

n n n

X X X X X

f d fd f f d f f d d A

= < =

Nh vy, lim .nn

X X

f d f d

=

Bi gii

( )a Ta bit, ( )xfnn lim tn ti khi v ch khi :

( ) ( )lim limn nf x f x= .

Nh vy, L = ( ) ( ){ }:lim limn nx f x f x=

Ta c nflim v nflim l nhng hm o c v nf o c .

t nn fff limlim = Ta c f l hm o c

Do , { }( )01= fL o c.

( )b Do ffea

n

.

nn ( ) 0: = CEFE v ffn trn E

Vi Ra , ta c ( ){ } ( ){ } ( ){ }axfExaxfExaxfXx C >>=> :::

( ){ } ( ){ }axfExaxfEx Cn >>= :lim:

Tp th nht v phi l tp o c , tp th hai l tp con ca CE l o

c ( do )

Do ( ){ }axfXx > : l o c

Suy ra f o c.

Bi 9: Cho { }nf l dy hm o c . Chng minh rng :

( )a L = ( ){ }:lim nn

x X f x tn tai

o c.

( )b Nu ( ),, FX , vi l o , v .a enf f th f o c.


63/73

LUN VN TT NGHIP


Bi gii

,0> t ( ){ }.: = xfxA nn Ta c : ( )0n

n n n

A A

A f d f d

V 0 dfA

n nn ( ) 0nA khi n

Vy, .0nf

Bi gii

V ffn nn { }nn ff k tha ff

eank

. trn .

Do ,peap

n ffk .

trn .

p dng b Fatou, ta c,

( )lim lim 1k kp p p

n n

R R R

f d f d f d =

Vy, .Bf

Bi gii

t n nA

a f d= v naa lim=

Khi , tn ti dy aaakk nn

: khi k

Bi 10: Cho dy hm o c khng m { }nf trn A tha mn: .0lim = dfA

n

Chng minh rng: .0nf

Bi 11: Vi 0>p , t =B { RRf : tha fo c v 1 dfp

R}.

Gi s rng { } Bf nn v .ffn Chng minh rng: .Bf

Bi 12: Cho dy hm o c { }nf ,f l nhng hm kh tch Lebesgue. Chng

minh rng: nu ffn trn A th .lim dfdf

A

n

A


64/73

LUN VN TT NGHIP


Do ffn nn

knf f khi k

V nh vy, tn ti dy con { } { } .:k k km ma e

n n nkmf f f f

Theo b Fatou, ta c:

lim lim limk k km m m

n n nm m m

A A A

f d f d f d a a

= = =

Nh vy, .lim dfdfA

n

A

Bi gii

Bi v ( ) ( )xgxfn hu khp ni v ( )g x kh tch trn nn nf kh tch trn .

p dng b Fatou, ta c:

lim nR R R

f d f d gd <

Suy ra f kh tch Lebesgue trn .

Vy, theo nh l hi t b chn, ta c:

lim nn

R R

f d f d

=

Do vy, 0lim = dffR

nn

Vi 20 00, : nR

n f f d n n > < >

t ( ) ( ){ }>= xfxfRxB n: th ( )21

n

R

B f f d

< < =

t \A B= .

Bi 13: Cho nff, l cc hm o c trn tha.a e

nf f trn R v tn ti

hm g kh tch trn R tha ( ) ( )xgxfn hu khp ni. Chng minh rng ffn hu nhu.


65/73

LUN VN TT NGHIP


Trn A ta c ( ) ( ) 0nnxfxfn >

Suy ra ffn hu nhu trn .

Bi gii

Do ffn nn { } { } :

k kn n nnkf f f f khi k

V knf g , g l hm kh tch, nn theo nh l hi t b chn, ta c:lim

knkf f

= ( )

M lim limkn nk k

f f

= , kt hp vi ( ) ta c: lim nn f f = .

Bi gii

Theo nh ngha ca lim nf ta c: { } { }kn n nkf f : l im limkn nk f f =

Vkn

f f nn .:k k kj j

a en n nf f f f

Theo b Fatou ta c: lim limk kj j

n nf f f = = l im knk f = lim nf

Vy, lim nf f

Bi 16: Cho ( ) ,nn E

E < hi t trung bnh v hm f . Chng minh Ef = h.k.n

vi E l mt tp o c.

Bi 15: Cho 0nf v nf f . Chng minh rng:

limf f

Bi 14: Cho nf g , g l hm kh tch v nf f . Chng minh rng:

lim nf f=


66/73

LUN VN TT NGHIP


Bi gii

t

=

=j

njnEE

1U ta c E o c v EEn khi n

DonE

hi t trung bnh v hm f nn fnE

. T tn ti dy con

feaEkn

. .

Vy Ef = h.k.n.

Bi gii

a. Vi ,0> ta lun c:

( ) ( ){ } : :2 2n n n n

f g f g x f f x g g

+ +

( ) ( ){ } : :2 2n n n n

f g f g x f f x g g

+ + +

Cho n , ta c:

( ) ( ){ } 0n nf g f g + +

Vy, gfgf nn ++ .

b. Ta c:

( )i Nu nf f th, naf af

, a .

( )ii Nu nf f th 2 0nf

iu ny suy ra t{ } { }.2 = nn ff

( )iii .gfgfn

Bi 17: Cho { }nnf , { }nng l cc dy hm o c trn A , ffn , ggn

.

Chng minh rng:

( )a gfgf nn ++ .

( )b gfgf nn ..

nu ( ) +


67/73

LUN VN TT NGHIP


Chng minh:

Ta c: { }=>In

ng .

Do ( )A < nn { }( ) 0g n >

Vi 0, > v 0 > tn ti 1n sao cho { }( ) 21

ng , chn 0n sao cho:

01 2

nn n f f n

=+22

Vy, .gfgfn

( )iv Ta chng minh ffn th 22 ffn

.

V ffn

nn .0

ffn

Do , theo( )ii ta c:

( )2

0nf f

Theo ( )i , ( )iii , v ( )a ta c:

( )22 2 2 2 0n n nf f f f f f f

= +

Vy, 22 ffn

( )v Sau cng, p dng cc bc trn v kt qu t cu a, ta c:

( ) ( )2 22 2 2 21 1 1 1 1 1

2 2 2 2 2 2n n n n n nf g f g f g f g f g fg= + + =

Vy gfgf nn .


68/73

LUN VN TT NGHIP


Bi gii

( ) ( )iii do nh ngha hi t theo o.

( )ii ( )iii do { } { }. > ffff nn

( ) ( )iiiii ,0,0 >> nu

,

2

Nn th :

{ }( ) ffn

>

2

ffn < .

( ) ( )ivii Khi . =

( ) ( )viv Do { } { }. > ffff nn

( ) ( )iiv ,0,0 >> chon

=

,

2min .

Khi , nu n P th:

{ }( ) ffn

> 2

ffn { }( )nf f > .

Bi 18: Cho ffn , l nhng hm o c trn A . Chng minh rng cc mnh

sau l tng ng.

( )i .ffn

( )ii ,:,0,0 NnN >> th { }( ) . ffn

( )iii ,0, 0, :N n N > > th { }( ) . > ffn

( )iv PnP > :,0 th { }( ) . ffn

( )v 0, :P n P > th { }( ) . > ffn

Bi 19: Gi { }nnE l dy nhng tp con o c ca [ ]ba, . Chng minh rng:

( )a Chng minh rngnE

hi tu v 0 trn [ ]ba, nu v ch nu =nE

vi n ln.

( )b Chng minh rng 0nE

trn [ ]ba, nu v ch nu ( ) = nn

Elim 0.


69/73

LUN VN TT NGHIP


Bi gii

( )a ( ) Vi ,2

1= chn 0N sao cho 0Nn > th 21 .

( ) Nu = nEN :0 , 0Nn > th ( ) 0=xnE . Do 00 0,nE n N = >

Vy,nE

hi tu v 0 trn [ ]ba, .

( )b Vi 10 = . Khi , ( )rBk o c v ( )( ) A

nk ffrrB

1

Vi ( ) ( )U=

=nk

kn rBrT , t gi thit ta c:

( )( )

=nk Akn ffr

rT1

0 khi 0n

Vy, ff uak . .

Bi gii

Chn 1 2,

.nn n

f n

=

Bi 20: Gi s{ } ff kk , l nhng hm kh tch trn A tha mn


70/73

LUN VN TT NGHIP


Vi 0 > , chn2

:N N

<

t2

0,AN

=

Ta c ( ) 2AN

= <

Mt khc, Cx A , ta c: 2 2 ,x n N N n

> >

Do , n N > v Cx A th ( ) 0nf x =

Vy, nf hi thu nhu v hm 0f = .

Tuy nhin,

1 2,

1. . 1n

n n

f d n nn

= = =

Do , 0nf f khi n

Vy, nf khng hi t trung bnh v f .

Bi gii

Do l o -hu hn nn tn ti cc tp , 1, 2,...jA j = sao cho:

1j

j

X A

=

= U , v ( ) ,jA j <

Khi , vi mi j , tn ti j jE A sao cho:

( )1

\j jA Ej

< , v nf fI trn jE

Ta c:

( )( )1 1

\ \j j j jj j

E A A E

= =

=U U

Bi 22: Cho l o -hu hn, .a enf f . Chng minh rng tn ti cc

tp 1 2, ,...E E trong X sao cho 1 0

C

jj E

=

= U vnf fI trn ,jE j


71/73

LUN VN TT NGHIP


( )( )1

\ \j jj

X A E

=

= U

( )1

\ \j jj

X A E

=

= I

Do :

( ) ( )1 1

1\ \

C

j j j j jj j

E A E A E j

= =

= <


72/73

LUN VN TT NGHIP


KT LUN

Nh vy, i vi dy hm o c c nhng dng hi khc nhau nh: hi t

theo o, hi t hu khp ni, hi t hu nhu, hi t trung bnh, V gia cc

dng hi t ny c mi lin h vi nhau. Cc mi lin h ny c s thay i khi

chng c xt trong khng gian o hu hn.

Tuy nhin, c th lun vn cha kp khai thc ht sa dng ca cc dng

hi t cng nh l mi lin h gia chng. Trong tng lai, nu c iu kin, em s

tip tc nghin cu su hn khai thc thm v ti ny.


73/73

LUN VN TT NGHIP

TI LIU THAM KHO

[1] Trn Th Thanh Thy, o v tch phn lebesgue, i hc Cn th,2007.

[2] u Th Cp,o v tch phn, NXB Gio dc, 2007.[3] Robert G. Bartle, A Modern Theory of Integration, American

Mathematical Society, 2001.[4] Robert G. Bartle, Solution Manual to A Modern Theory of Integration,

American Mathematical Society, 2001.[5] Shmuel Kantorovitz, Introduction to Modern Analysis, Oxford

Mathematics, 2004.[6] Avner Friedman, Foundations of Modern Analysis, Dover Publications,

2001.

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