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Chapter 12
Borel and Radon Measures on theReal Line
Chapter 5 presented the theory of Lebesgue measure and
integration in theEuclidean spaces Rd. Measure theory can be
developed on an abstract mea-sure space X, but in many contexts in
analysis we deal with measures on atopological space X. On a
topological space our measure should probably bedefined on all of
the open sets, closed sets, countable intersections of opensets,
countable unions of closed sets and so forth. This leads to the
idea of aBorel measure on X, and the slightly more restrictive
notion of Radon mea-sures. Not surprisingly, we need to impose some
conditions on X beyond themere existence of a topology; usually we
require that X be a locally compactHausdorff space (LCHS). This
chapter is an introduction to the theory ofsigned and complex Borel
measures and Radon measures. In the spirit of anintroduction, we
take X = R. This is one of the most common settings whereBorel and
Radon measures are encountered. Moreover, it yields a good in-sight
into the properties of Borel and Radon measures on LCHS but
withoutsome of the technical complications that arise when
considering completelygeneral LCHS. For more details on Borel and
Radon measures, especially forabstract LCHS, we refer to Folland
[Fol99].
12.1 -Algebras
The Axiom of Choice implies that there is no way to create a
function defined on every subset of R so that all of the following
hold:
(i) 0 (E) for every E R,
(ii) ([a, b]) = b a,
(iii) if E1, E2, . . . are finitely or countably many disjoint
subsets of R, then(kEk) =
k (Ek),
(iv) (E + h) = (E) for all E R and h R.
c2012 Christopher Heil425
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426 12 Borel and Radon Measures on the Real Line
There are several ways to address this. In Chapter 5 we began
withLebesgue exterior measure | |e, which satisfies (i), (ii), and
(iv), but failsrequirement (iii). Rather unsettlingly, for Lebesgue
exterior measure the as-sumption E F = does not imply |E F |e =
|E|e + |F |e.
In order to obtain Lebesgue measure | |, we therefore dropped
require-ment (i), with the result that not all subsets of R are
Lebesgue measurable.Although we can no longer measure every set, we
do have the satisfying factthat requirements (ii), (iii), and (iv)
hold for all those sets E R that areLebesgue measurable.
On the other hand, there are good reasons for relaxing the
requirementsin other ways. For example, one of the most important
measures is the measure, which assigns the size 1 or 0 to a set E
depending on whether theorigin belongs to E or not. Requirements
(ii) and (iv) are not satisfied bythe measure, but both (i) and
(iii) do hold. Other alternatives are to allowa measure to take
real or complex values, instead of just nonnegative values.This
leads to the idea of signed and complex measures on R.
In this chapter we will present the definitions and properties
of abstractBorel and Radon measures on the real line. In order to
give a useful definitionof a measure, we must first decide on the
properties that a class of sets shouldpossess in order to be
measured.
Definition 12.1.1 (-Algebra). A -algebra on R is a nonempty
collection of subsets of R which satisfies:
(a) is closed under both finite and countable unions:
E1, E2, =k
Ek ,
(b) is closed under complements:
E = EC = R\E .
If is a -algebra then it is nonempty and therefore contains some
set E R. Hence also contains R\E, and therefore contains both R = E
(R\E)and = R\R.
We saw in Chapter 5 that the class L of Lebesgue measurable
subsets ofR forms a -algebra. The power set P(R) = {E : E R} is
trivially another-algebra. At the other extreme, {,R} is a
-algebra.
Given a particular class E of subsets of R, there will be many
-algebrasthat contain E . However, there is a unique smallest
-algebra that contains E .
Exercise 12.1.2. Let E be a nonempty collection of subsets of R.
Show that
(E) ={
: is a -algebra and E }
is a -algebra on R. We call (E) the -algebra generated by E
.
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12.2 Signed Measures 427
Note that if 1, 2 are -algebras, then 1 2 is not formed by
in-tersecting the elements of 1 with those of 2. Rather, it is the
collectionof all sets that are common to both 1 and 2. Thus, if is
a -algebrathat contains E then (E) , which explains why (E) is the
smallest-algebra that contains E .
The next definition introduces the particular -algebra that will
concernus in this chapter.
Definition 12.1.3 (Borel -algebra). The Borel -algebra B on R is
thesmallest -algebra that contains all the open subsets of R. That
is,
B = (U) where U = {U R : U is open}.
The elements of B are called the Borel subsets of R.
In particular, B includes all the open and closed subsets of R,
as well asthe G and F sets that were introduced in Definition
5.2.18. However, notevery subset of R is a Borel set. The Borel
-algebra can be defined on Rd
or Cd in an analogous manner.Although our focus will be on the
Borel -algebra on R, many of the
definitions and results that we will discuss are valid on more
general domains.However, it is often instructive to consider the
even simpler case of measureson the natural numbers N. Some of the
additional problems at the end of eachsection deal with this
setting. The topology on N is the discrete topology, i.e.,every
subset of N is open, so every subset of N is a Borel set. In other
words,the Borel -algebra on N is P(N), the power set of N. Whenever
we speak ofa measure on N, we will assume that the associated
-algebra is P(N).
12.2 Signed Measures
Definition 12.2.1 (Signed Measure). A function : B [,] is
asigned Borel measure on R, or simply a signed measure, if
(a) () = 0,
(b) takes at most one of the values ,
(c) if E1, E2, . . . are finitely or countably many disjoint
Borel sets, then
(k
Ek
)=k
(Ek).
If (E) 0 for each E B, then we say that is a positive measure,
andin this case we write 0.
If |(E)| < for each E B, then we say that is a bounded
measureor a finite measure.
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428 12 Borel and Radon Measures on the Real Line
If |(K)|
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12.2 Signed Measures 429
Note that a is not translation-invariant, as a(E) and a(E + h)
are notequal in general. Another difference from Lebesgue measure
is that the set{a}, which Lebesgue measure regards as an
insignificant zero measure set,has measure 1 with respect to a.
Here is another example of an unbounded positive measure, very
differentfrom Lebesgue measure in many ways.
Exercise 12.2.6. Define (E) to be the cardinality of E if E is a
finite set,and otherwise. Show that is a positive, unbounded Borel
measure thatis not locally finite. We call counting measure on
R.
Only finite sets have finite measure with respect to counting
measure,while every finite set has Lebesgue measure zero.
We can create signed measures from positive measures.
Exercise 12.2.7. Show that if 1, 2 are positive measures with at
least oneof 1, 2 bounded, then 1 2 is a signed measure.
A positive measure has the useful property of monotonicity: if
A, Bare Borel sets and A B, then (A) (B). In particular, if is
positiveand (E) = 0, then (A) = 0 for every Borel A E. However, for
a signedmeasure it is important to distinguish between sets E that
satisfy (E) = 0and sets that are null for in the following
sense.
Definition 12.2.8 (Null Sets). We say that a signed measure is
null ona Borel set E B if (A) = 0 for every A B with A E.
Definition 12.2.9 (Mutually Singular Measures). Two signed
measures, are mutually singular, denoted , if there exist E, F B
such that
(a) E F = R and E F = ,
(b) is null on F, and
(c) is null on E.
Exercise 12.2.10. Show that Lebesgue measure and the measure a
are mu-tually singular, i.e., dx a.
12.2.1 The Jordan Decomposition
Now we come to a fundamental decomposition for signed measures.
The fol-lowing exercise motivates this by considering the special
case of measures ofthe form (E) =
Ef(x) dx.
Exercise 12.2.11. Let (E) =Ef(x) dx be a measure of the form
con-
structed in Exercise 12.2.4. Set P = {f 0} and N = {f < 0}.
By chang-ing P and N by a set of measure zero, we may assume that P
and N areBorel sets (see Exercise 5.2.21). For E B define
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430 12 Borel and Radon Measures on the Real Line
+(E) = (E P ) =
E
f+(x) dx,
(E) = (E N) =
E
f(x) dx.
Show that +, are positive measures, = + , and + .
Although we will not prove it, the next result states that this
same kindof decomposition holds for arbitrary signed measures.
Theorem 12.2.12 (Jordan Decomposition Theorem). If is a
signedBorel measure on R, then there exist unique positive Borel
measures +,
such that = + and + .
Consequently, by definition of mutually singular measures, given
a signedmeasure there exist disjoint Borel sets P, N R with P N = R
such that = + , is null on P, and + is null on N. We call = +
theJordan decomposition of (since it is unique), and R = P N an
associatedHahn decomposition of R (it is not unique).
Definition 12.2.13 (Positive, Negative, and Total Variation
Mea-sures). Given a signed Borel measure , let = + be its
Jordandecomposition.
(a) We call + the positive variation of , and the negative
variation of .
(b) The total variation of is the positive measure
|| = + + .
That is, || is defined by
||(E) = +(E) + (E), E B. (12.1)
Observe that equation (12.1) implies that
|(E)| ||(E), E B.
Further, since || is a positive measure, it is monotonic, and
hence
||(E) ||(R), E B.
Of course, ||(R) could be infinite, but if it should be finite
then it follows that|(E)| is bounded by the finite quantity ||(R)
for every E B, and hence is a bounded measure. In fact, the next
exercise shows that the converse isalso true, which explains the
terminology bounded measure instead of justfinite measure: If (E)
is finite for all Borel sets E, then there is a finiteupper bound
to the values of |(E)|.
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12.2 Signed Measures 431
Exercise 12.2.14. Let be a signed Borel measure. Prove that
is bounded ||(R)
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432 12 Borel and Radon Measures on the Real Line
Here are some useful equivalent formulations of the positive,
negative, andtotal variation measures.
Exercise 12.2.15. Let be a signed Borel measure. Show that if E
B,then
+(E) = sup{(A) : A B, A E
},
(E) = inf{(A) : A B, A E
},
||(E) = sup
{ nk=1
|(Ek)| : n N, Ek B, E =n
k=1
Ek disjointly
}.
Now that we have defined the total variation measure, we can
define -finite measures.
Definition 12.2.16 (-Finite Measures). Let be a signed Borel
mea-sure. If we can write R = Ek using at most countably many sets
Ek Beach with ||(Ek)
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12.3 Positive Measures and Integration 433
12.3 Positive Measures and Integration
The next few sections are devoted to developing the theory of
integration withrespect to signed measures, beginning in this
section with positive measures.As the theory of integration with
respect to positive measures very closelyparallels the theory of
integration with respect to Lebesgue measure that waspresented in
Section 6.4, we shall be brief and simply state the main resultsof
this section without proof.
12.3.1 Basic Properties of Positive Measures
Theorem 12.3.1. Let be a positive Borel measure on R.
(a) Monotonicity: If A, B B and A B, then (A) (B).
(b) If A, B B, B A, and (B)
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434 12 Borel and Radon Measures on the Real Line
(b) A function f : R [,] is Borel measurable if
E [,] and E R is a Borel set
= f1(E) R is a Borel set.
Because the Borel -algebra is generated by the open sets, a
functionf : R C is Borel measurable if f1(U) is a Borel set for
each open setU C. If f is real-valued, then f is Borel measurable
if f1(a,) is a Borelset for every a R, and hence the definition of
Borel measurable functions isentirely analogous to the definition
of Lebesgue measurable functions givenin Definition 6.1.1. In
particular, every continuous function on R is Borelmeasurable.
Lemma 12.3.3. (a) If f, g : R R are Borel measurable, then so
are f + gand fg.
(b) If fn : R R are Borel measurable for n N, then so are sup
fn, inf fn,lim sup fn, and lim inf fn. Consequently, if f(x) = limn
fn(x) existsfor each x, then f is Borel measurable.
Appropriate parts of Lemma 12.3.3 extend to complex-valued
functions,and can also be extended to extended real-valued
functions if we are carefulabout instances where we encounter .
Compositions of Borel measurable functions also behave well. If
we havetwo Borel measurable functions f, g : R R, then it follows
directly fromthe definition that g f is also Borel measurable.
Generalizing the definitionof Borel measurability in the natural
way to functions on C, if f : R C andg : C C are Borel measurable,
then so is g f. In particular, f2, |f |, |f |p,etc., are all Borel
measurable if f is.
12.3.3 Integration of Nonnegative Functions
Simple functions are defined just as in Definition 6.4.1, except
that now werequire our functions to be Borel measurable instead of
Lebesgue measurable.Thus, a simple function on the real line is a
Borel measurable function on Rthat assumes only finitely many
distinct scalar values. If these distinct valuesare a1, . . . , aN
and we let Ek be the set where takes the value ak (that is,
Ek = { = ak}), then =N
k=1 akEk is the standard representation of .
Definition 12.3.4 (Integral of Nonnegative Functions). Let be a
pos-itive Borel measure on R.
(a) If 0 is a simple function on R with standard representation
=Nk=1 akEk , then the integral of with respect to is
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12.3 Positive Measures and Integration 435d =
(x) d(x) =
Nk=1
ak (Ek).
(b) If f : R [0,] is Borel measurable, then the integral of f
with respectto is
f d =
f(x) d(x) = sup
{d : 0 f, simple
}.
We writeEf d to mean
f E d.
If is a positive measure and a certain property holds except for
a set Ewith (E) = 0, then we say that this property holds -a.e.
We have the following convergence theorem for positive measures,
analo-gous to Theorem 6.4.13 for Lebesgue measure.
Theorem 12.3.5 (Monotone Convergence Theorem). Let be a
pos-itive Borel measure on R, and let {fn}nN be a sequence of Borel
mea-surable, nonnegative, monotone increasing functions on R. If we
definef(x) = limn fn(x), then
limn
fn d =
f d.
Corollary 12.3.6. Let be a positive Borel measure on R. If
{fn}nN be asequence of Borel measurable, nonnegative functions on
R, then (
n=1
fn
)d =
n=1
fn d.
As in Theorem 6.4.6, we can always create a sequence of simple
functionsthat increases monotonically to a given nonnegative f.
Combining this withthe Monotone Convergence Theorem, we obtain the
following facts.
Theorem 12.3.7. The following properties hold for any positive
Borel mea-sure on R and any Borel measurable functions f, g : R
[0,].
(a)f d = 0 if and only if f = 0 -a.e.
(b) If f g -a.e., thenf d
g d.
(c)(f + g) d =
f d+
g d.
(d) If c 0 then(cf) d = c
f d.
(e) If A, B B and A B, thenAf d
Bf d.
Theorem 12.3.8 (Fatous Lemma). Let be a positive Borel measureon
R. If {fn}nN is a sequence of Borel measurable, nonnegative
functionson R, then (
lim infn
fn
)d lim inf
n
fn d.
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436 12 Borel and Radon Measures on the Real Line
12.3.4 Integration of Arbitrary Functions
Next we extend integration with respect to a positive measure to
arbitraryfunctions.
Definition 12.3.9 (Integrable Functions). Let be a positive
Borel mea-sure on R.
(a) A Borel measurable function f : R [,] is a (real) extended
-
integrable function if at least one off+ d,
f d is finite.
(b) A Borel measurable function f : R [,] or f : R C is
-integrableif|f | d is finite.
Definition 12.3.10 (Integration). Let be a positive Borel
measure onR.
(a) If f is a real, extended -integrable function, then we
definef d =
f+ d
f d.
(b) If f : R C is Borel measurable andRe(f) d and
Im(f) d both
exist and are finite, then we definef d =
Re(f) d+ i
Im(f) d.
In all other cases,f d is undefined. Note in particular that if
f is
complex-valued, thenf d, if it exists, is a complex scalar. On
the other
hand, if f is real-valued, thenf d, if it exists, can be either
a finite real
scalar or .
Lemma 12.3.11. If is a positive Borel measure on R andf d
exists,
then f d |f | d. When defining the space L1() of functions that
are integrable with respect
to , we have a choice between letting our functions be extended
real-valuedor complex-valued. In this volume, we will take L1() to
consist of complex-valued -integrable functions.
Definition 12.3.12. If is a positive Borel measure on R, then
L1() con-sists of -integrable functions f : R C. The L1-norm of f
L1() is
f1 =
|f | d.
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12.3 Positive Measures and Integration 437
There are many other notations commonly used to denote L1(),
includingL1(d), L1(R;), or L1(R; d).
Note the implicit dependence of the notation f1 on . When we
needto emphasize the dependence on , we will write f1, =
|f | d.
Theorem 12.3.13. If we identify functions in L1() that are equal
-a.e.,then 1 is a norm on L
1(), and L1() is complete with respect to thisnorm.
As for Lebesgue measure, the following result is one of the most
usefulconvergence theorems.
Theorem 12.3.14 (Dominated Convergence Theorem). Let be a
pos-itive Borel measure on R. Let {fn}nN be a sequence of Borel
measurablefunctions on R such that:
(a) fn(x) f(x) for -a.e. x, and
(b) there exists g L1() such that |fn(x)| g(x) -a.e. for every
n.
Then fn converges to f in L1-norm, i.e.,
limn
f fn1 = limn
|f fn| d = 0.
Consequently, limnfn d =
f d.
Problems
12.3.15. Show that if f is Borel measurable and a R, thenf da =
f(a).
Characterize L1(a).
12.3.16. Let be a positive Borel measure on R, and suppose that
g 0is Borel measurable. Show that (E) =
Eg d is a positive Borel measure,
and if f L1(), thenf d =
fg d.
12.3.17. Given a positive Borel measure on R, let S be the set
of all simplefunctions =
Nk=1 ckEk such that (Ek) < for each k. Show that S is
dense in L1().
12.3.18. Given a positive Borel measure on R, show that if fn f
inL1() then there exists a subsequence such that fnk f pointwise
a.e.
12.3.19. Let be a positive measure on N. Set w(k) = {k}. Show
that if f =
(f(k))kN is a nonnegative sequence of scalars, thenf d =
f(k)w(k).
Conclude that L1() = 1w, the weighted 1 space defined in Problem
1.3.17.
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438 12 Borel and Radon Measures on the Real Line
12.4 Signed Measures and Integration
We extend integration to signed measures by making use of the
Jordan de-composition of the measure.
Definition 12.4.1. Let be a signed Borel measure on R, and let =
+ be its Jordan decomposition. Assume that f is a Borel measurable
mapof R into either [,] or C. If
|f | d+,
|f | d
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12.4 Signed Measures and Integration 439
E B, (E) = 0 = (E) = 0.
Exercise 12.4.7. Show that the measures , defined in Exercise
12.4.4satisfy .
Now we come to a major structure result for -finite signed
measures.
Theorem 12.4.8 (LebesgueRadonNikodym Theorem). Let be a-finite
signed measure and let be a -finite positive measure on R.
(a) There exist unique -finite signed Borel measures , such
that
= + , , .
(b) There exists a real, extended -integrable function f such
that d = f d,i.e.,
= f d+ .
(c) If we also have = f d + where f is a real, extended
-integrable
function, then f = f -a.e.
We refer to = + as the Lebesgue decomposition of with respect
tothe measure .
Corollary 12.4.9 (RadonNikodym Theorem). If is a -finite
signedmeasure and is a -finite positive measure such that , then
thereexists a real, extended -integrable function f such that d = f
d. Any twofunctions which have this property are equal -a.e.
Definition 12.4.10 (RadonNikodymDerivative). The function f
givenin Corollary 12.4.9 is called the RadonNikodym derivative of
with respectto , often denoted f = d
d. With this notation we have d = d
dd. The
RadonNikodym derivative is unique up to sets of -measure
zero.
Note that if d = f d, then Exercise 12.4.4 implies that d|| = |f
| d.
Exercise 12.4.11. Let be a -finite signed measure, and let , be
-finitepositive measures. Show that
and = .
Further, if d = f d and d = g d, then d = fg d.
Remark 12.4.12. The measure is not absolutely continuous with
respect toLebesgue measure. Assuming a willing suspension of belief
for the moment,if we did have dx (which we do not), then there
would exist a function(x) such that d = (x) dx. By Problem 12.3.15
we know that
f d = f(0),
so this means that we would have
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440 12 Borel and Radon Measures on the Real Line
f(0) =
f d =
f(x) (x) dx.
Although there is no such function (x), it is common to abuse
notationand write
f(x) (x) dx = f(0), even though what is really meant with
these symbols is integration of f with respect to the measure,
which in ournotation should be written as
f d or
f(x) d(x).
The following exercise suggests why the terminology absolute
continuityis used in connection with the relation .
Exercise 12.4.13. Let be a bounded signed Borel measure and a
positiveBorel measure on R. Prove that if and only if
> 0, > 0 such that E B, (E) < = |(E)| < .
Problems
12.4.14. Let = + be the Jordan decomposition of a signed
Borelmeasure , and let R = P N be an associated Hahn decomposition.
Showthat d+ = P d, d
= N d, and d|| = (P N ) d.
12.4.15. Show that if is a signed Borel measure, then
||(E) = sup
{E
f d
: |f | 1}.12.4.16. Show that if is a signed Borel measure and a
positive Borelmeasure such that and , then = 0.
12.4.17. Given a signed Borel measure and a positive Borel
measure ,show that
|| +, .
12.4.18. Let denote counting measure on R (see Exercise
12.2.6).
(a) Prove that if f : R [0,] is Borel measurable, thenf d =
sup
{ Nj=1
f(xj) : N N, xj R
}.
(b) Prove that dx , but dx 6= f d for any function f.
(c) Prove that has no Lebesgue decomposition with respect to dx,
i.e., theredo not exist signed measures and such that = + , dx, and
dx.
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12.5 Complex Measures 441
12.5 Complex Measures
Next we expand the class of measures by allowing them to be
complex-valued.
Definition 12.5.1 (Complex Measure). A function : B C is a
com-plex Borel measure on R, or simply a complex measure, if
(a) () = 0,
(b) if E1, E2, . . . are finitely or countably many disjoint
Borel sets, then
(k
Ek
)=k
(Ek).
Note that for a complex measure we have |(E)| < for every
Borelset E. The following exercise shows that a complex measure is
bounded.
Exercise 12.5.2. Let be a complex Borel measure. For E B,
definer(E) = Re((E)) and i(E) = Im((E)). Show that r, i are
boundedsigned measures, and for any E B we have
|(E)| |r(E)| + |i(E)| |r|(R) + |i|(R).
Conclude that is bounded in the sense that supEB |(E)|
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442 12 Borel and Radon Measures on the Real Line
Next we give an important structure theorem for complex measures
(com-pare Theorem 12.4.8 for the case of signed measures).
Theorem 12.5.7 (LebesgueRadonNikodym Theorem). Let be acomplex
Borel measure on R, and let be a -finite positive measure on R.Then
there exists a function f L1() and a complex Borel measure
suchthat
= f d+ , . (12.3)
If we also have = f d + where f L1(d) and , then = and
f = f -a.e.
We will need the following exercise in order to define the total
variationof a complex measure.
Exercise 12.5.8. Let be a complex Borel measure on R, and define
=|r| + |i|, so is a positive bounded Borel measure. Show there
exists afunction f L1() such that d = f d.
The total variation of a complex measure is a little more
awkward to definethan it is for a signed measure. By Exercise
12.5.8, we know that if is acomplex measure, then there exists at
least one positive measure and onefunction f L1() such that d = f
d. We will define the total variation of to be the measure d|| = |f
| d, but of course we need to know that this iswell-defined. The
following theorem shows that this definition is independentof the
choice of and f.
Theorem 12.5.9. Let be a complex Borel measure on R. If 1, 2
arebounded positive measures and f1 L
1(1), f2 L1(2) are such that
f1 d1 = d = f2 d2, then |f1| d1 = |f2| d2.
Proof. Define = 1 + 2. Then since 1 , there exists a functiong1
L
1() such that d1 = g1 d. Likewise, there exists some g2 L1()
such that d2 = g2 d. Because 1, 2, 0, we have g1, g2 0
-a.e.Thus, we have d = f1d1 and d1 = g1 d. Exercise: Show that
Exercise
12.4.11 generalizes to complex measures, and use this to show
that d =f1g1 d an d = f2g2 d (see also Problem 12.3.16).
The uniqueness statement in the LebesgueRadonNikodym
Theoremtherefore implies that f1g1 = f2g2 -a.e. Consequently,
|f1| g1 = |f1g1| = |f2g2| = |f2| g2 -a.e.,
and hence
|f1| d1 = |f1| g1 d = |f2| g2 d = |f2| d2.
Definition 12.5.10 (Total Variation of a Complex Measure). Let
be a complex Borel measure on R. Then the total variation || of is
the
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12.5 Complex Measures 443
positive measure d|| = |f | d, where is any positive measure and
f is anyfunction in L1() such that d = f d.
Next we give some properties of complex measures.
Exercise 12.5.11. Let be a complex Borel measure on R. Show that
thefollowing statements hold.
(a) |(E)| ||(E) for all E B.
(b) ||, and there exists a function g such that |g| = 1 ||-a.e.
andd = g d||.
(c) If f L1(), then f d |f | d||.
The representation d = g d|| in part (b) of the preceding
exercise iscalled the polar decomposition of .
The following equivalent reformulations of the total variation
measure areoften easier to employ in practice than Definition
12.5.10.
Exercise 12.5.12. Let be a complex Borel measure on R. Prove the
fol-lowing equivalent characterizations of ||.
(a) ||(E) = sup
{ nk=1
|(Ek)| : n N, Ek B, E =n
k=1
Ek disjointly
}.
(b) ||(E) = sup
{ k=1
|(Ek)| : Ek B, E =k=1
Ek disjointly
}.
(c) ||(E) = sup
{E
f d
: |f | 1 ||-a.e.}. Exercise 12.5.13. Show that if is a complex
measure, then E B is anull set for if and only if ||(E) = 0.
For a complex measure , we say that a property holds -almost
everywhereif it holds except on a null set for . Thus -almost
everywhere is the sameas ||-almost everywhere.
The space Mb(R) of all complex Borel measures is a Banach
space.
Definition 12.5.14 (Space of Complex Borel Measures). We set
Mb(R) ={ : is a complex Borel measure on R
},
and define the norm of a complex measure to be
= ||(R). (12.4)
Exercise 12.5.15. Show that as defined in equation (12.4) is a
norm onMb(R), and Mb(R) is a Banach space with respect to this
norm.
We identify some particular subspaces of Mb(R).
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444 12 Borel and Radon Measures on the Real Line
Exercise 12.5.16. Show that if be a positive Borel measure on R
andd = f d where f L1(), then = f1 =
|f | d.
Consequently, if is a positive measure, then L1() Mb(R). More
pre-cisely, if we define dg = g d for each g L
1(), then Exercise 12.5.16shows that T : g 7 g is an isometric
embedding of L
1() into Mb(R). If is -finite, then range(T) = { Mb(R) : }. In
particular, Lebesguemeasure is a positive Borel measure, and hence
if we identify f L1(R) withf dx Mb(R), then L
1(R) Mb(R).
Definition 12.5.17. The space of bounded discrete Borel measures
is
Md(R) =
{ Mb(R) : =
k=1
ckak , distinct ak R,
k=1
|ck|
-
12.6 Fubini and Tonelli for Borel Measures 445
12.5.25. Given a complex measure on N, find an explicit
description of ||.
12.5.26. Show that 7 ({k})kN is an isometric isomorphism of
Mb(N)
onto 1(N). Thus Mb(N) = 1(N). Compare Exercise 9.5.11.
12.6 Fubini and Tonelli for Borel Measures
In this section we will state Fubinis and Tonellis theorems for
Borel mea-sures. The definition of measurability on R2 is analogous
to the definition forR.
Definition 12.6.1. (a) The Borel -algebra on R2 is the smallest
-algebraof subsets of R2 that contains all the open subsets of
R2.
(b) A function F : R2 C is Borel measurable if
E C is a Borel set = F1(E) R2 is a Borel set.
(c) A function F : R2 [,] is Borel measurable if
E [,] and E R is a Borel set
= F1(E) R2 is a Borel set.
Tonellis and Fubinis Theorems apply to all -finite positive
measures.
Theorem 12.6.2 (Tonellis Theorem). Let , be -finite positive
Borelmeasures on R. If F : R2 [0,] is Borel measurable, then the
followingstatements hold.
(a) Fx(y) = F (x, y) is Borel measurable for every x R.
(b) F y(x) = F (x, y) is Borel measurable for every y R.
(c) g(x) =Fx(y) d(y) is Borel measurable.
(d) h(y) =F y(x) d(x) is Borel measurable.
(e)
(F (x, y) d(x)
)d(y) =
(F (x, y) d(y)
)d(x).
Theorem 12.6.3 (Fubinis Theorem). Let , be -finite positive
mea-sures on R. If F : R2 [,] or F : R2 C is Borel measurable
and
|F (x, y)| d(x) d(y) < , (12.5)
then the following statements hold.
(a) Fx(y) = F (x, y) is Borel measurable and -integrable for
-a.e. x R.
(b) F y(x) = F (x, y) is Borel measurable and -integrable for
-a.e. y R.
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446 12 Borel and Radon Measures on the Real Line
(c) g(x) =Fx(y) d(y) is Borel measurable and -integrable.
(d) h(y) =F y(x) d(x) is Borel measurable and -integrable.
(e)
(F (x, y) d(x)
)d(y) =
(f(x, y) d(y)
)d(x).
Although stated for positive measures, Fubinis Theorem extends
to signedand complex measures. Suppose that , are -finite signed
measures andF : R2 [,] is a Borel measurable function that
satisfies
|F (x, y)| d||(x) d||(y) < . (12.6)
Then by breaking and into positive and negative parts and
applyingFubinis theorem to each of those parts, we see that the
conclusions of FubinisTheorem hold for F. Likewise, if , are
complex measures (hence bounded)and F : R2 C is measurable, by
breaking into real and imaginary partswe again see that the
conclusions of Fubinis Theorem hold if F satisfiesequation
(12.6).
12.7 Radon Measures
Now we introduce Radon measures on the real line. Radon measures
can bedefined on any locally compact Hausdorff space, but because
we are onlydealing with the real line, certain simplifications
occur. Most of these are dueto the fact that R is -compact, i.e.,
it can be written as a countable unionof compact sets.
Definition 12.7.1 (Radon Measures). Let be a positive Borel
measureon R.
(a) is outer regular on E B if (E) = inf{(U) : U E, U open
}.
(b) is inner regular on E B if (E) = sup{(K) : K E, K
compact
}.
(c) If is both inner and outer regular on every Borel set, then
is regular.
(d) is locally finite if (K)
-
12.7 Radon Measures 447
only if is a locally finite positive Borel measure on R.
However, on domainsother than Euclidean space, the distinction
becomes more important, and werefer to [Fol99] for complete
details.
The first step in showing the equivalence of Radon measures with
locallyfinite Borel measures is the following result.
Theorem 12.7.2. Every Radon measure on R is regular.
Proof. By definition, a Radon measure is outer regular, so we
just have toshow that it is inner regular on every Borel set.
Suppose first that E is a Borel set with (E) 0.Since is outer
regular on E, there exists an open set U E such that(E) (U) <
(E) + . As U is open and is inner regular on open sets,there exists
a compact set F U such that (F ) > (U) .
Now, since E has finite measure, (U\E) = (U) (E) < . Also, is
outer regular on U\E, so there exists an open set V U\E such that(V
) < .
Set K = F\V. Then K is compact, K E, and
(K) = (F ) (F V ) > (U) (V ) > (E) 2.
Hence is inner regular on any Borel set E that has finite
measure.Now suppose that E is a Borel set with (E) =. Define Ek =
E[k, k].
Then, since is locally finite, (Ek) is finite. Further, E1 E2
andE = Ek, so limk (Ek) = (E) = by Theorem 12.3.1. Hence givenR
> 0, there exists a k such that (Ek) > R. Since is inner
regular on Ek,there exists a compact set K Ek such that (K) > R.
Hence
sup{(K) : K E, K compact
}= = (E),
so is inner regular on E.
The fact that the real line is -compact, i.e., is a union of
countably manycompact sets, is clearly an important ingredient of
the preceding proof. Ona general space, a Radon measure will be
inner regular on any subset that is-finite.
We state the following useful property of Radon measures without
proof.
Theorem 12.7.3 (Luzins Theorem). Let be a Radon measure on R.
Iff : R C is Borel measurable and
({f 6= 0}
)< , then for every > 0,
there exists a function Cc(R) such that
({f 6= }
)< .
Further, if f is bounded, then can be chosen so that
supxR
|(x)| supxR
|f(x)|.
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448 12 Borel and Radon Measures on the Real Line
Problems
12.7.4. Show that if is a Radon measure, then Cc(R) is a dense
subspaceof L1().
12.7.5. Since every subset of N is open, every positive measure
on N isa Borel measure. Show that is locally finite if and only if
{k} < forall k N, and the Radon measures on N are precisely the
locally finitepositive measures on N.
12.8 The Riesz Representation Theorem for PositiveFunctionals on
Cc(R)
In this section we will discuss one form of the Riesz
Representation Theo-rem. This version proves an equivalence between
Radon measures and positivelinear functionals on Cc(R). While this
theorem is only concerned with posi-tive measures and positive
functionals, in Section 12.10 we will see a secondRiesz
Representation Theorem that deals with complex Radon measures
andbounded functionals.
In this section we deal both with measures and functionals.
Typically, wewill let denote a functional and a measure. In keeping
with Notation 9.5.5,we write f, to denote the action of a linear
functional : Cc(R) C ona vector f Cc(R). Further, f, is a
sesquilinear form, linear in f butantilinear in .
Each (positive) Radon measure on R induces an associated linear
func-tional on Cc(R) by the formula
f, =
f d, f Cc(R). (12.7)
This example immediately raises several questions, which we will
address inthis section. First, is the functional defined in
equation (12.7) continuous onCc(R)? Of course, continuity is not
even defined until we specify the topologyon Cc(R), and, as it
turns out, there is more than one natural choice.
Second, once we specify the topology on Cc(R), does every
continuouslinear functional on Cc(R) have the form given in
equation (12.7)? In otherwords, can we characterize the dual space
of Cc(R)? This question also re-quires some refinement, since we
have specified that Radon measures arepositive measures, whereas if
we let be a complex measure then we can stilldefine a functional by
equation (12.7).
To address these questions, we will discuss two particular
topologies onCc(R).
-
12.8 The Riesz Representation Theorem for Positive Functionals
on Cc(R) 449
12.8.1 Topologies on Cc(R)
Since we wish to study the continuity of linear functionals on
Cc(R), wemust specify a topology or a convergence criterion on
Cc(R). The followingexamples give two natural choices.
Example 12.8.1 (The Uniform Topology). Cc(R) is a normed space
with re-spect to the topology induced by the uniform, or L, norm. A
linear func-tional on Cc(R) is continuous with respect to the
uniform topology if andonly if it is bounded with respect to the L
norm. That is, is continuousif and only if there exists a constant
C > 0 such that
|f, | C f, all f Cc(R). (12.8)
Since Cc(R) is a dense subspace of the Banach space C0(R),
Exercise 9.1.18implies that such a has a unique extension to a
bounded linear functionalon all of C0(R), and we also refer to this
extension as .
Example 12.8.2 (The Inductive Limit Topology). For each compact
set K R, define
C(K) ={f Cc(R) : supp(f) K
}.
Each of the spaces C(K) is a Banach space with respect to the
uniform norm.Further, as a set,
Cc(R) ={
C(K) : K R, K compact}.
We can define a topology on Cc(R) by declaring that a function f
is continu-ous on Cc(R) if and only if for each compact K the
restriction of f to C(K)is continuous with respect to the L-norm on
C(K).
In particular, a linear functional : Cc(R) C is continuous with
respectto this topology if and only if |C(K) : C(K) C is continuous
for eachcompact set K. Since C(K) is a normed space, this happens
if and only ifeach restriction |C(K) is bounded with respect to the
norm on C(K), whichmeans that there exists a constant CK > 0
such that
|f, | CK f, all f C(K). (12.9)
However, unlike the boundedness statement with respect to the
uniformtopology given by equation (12.8), which has a single
constant C, the con-stants CK in equation (12.9) can depend on the
compact set K.
In technical language, this topology on Cc(R) is the inductive
limit of thetopologies (C(K), ) over compact K, and hence we will
refer to it asthe inductive limit topology on Cc(R). This type of
topology is also discussedin Section 14.6, and we refer to [Con90]
or [Rud91] for definition of the opensets and complete details on
inductive limit of topologies.
-
450 12 Borel and Radon Measures on the Real Line
The following definition makes precise the convergence criterion
on Cc(R)corresponding to each of these two topologies.
Definition 12.8.3. Let {fn} be a sequence of functions in
Cc(R).
(a) We say that fn converges to f uniformly if f fn 0 as n.
Inthis case, we write fn f uniformly.
(b) We say that fn converges to f in Cc(R) if there exists a
compact set Ksuch that supp(fn) K for all n, and f fn 0 as n. In
thiscase, we write fn f in Cc(R).
Note that
fn f in Cc(R) = fn f uniformly . (12.10)
However, the converse implication does not hold in general, so
these aretwo distinct topologies on Cc(R). Equation (12.10) implies
that the uniformtopology on Cc(R) is weaker than the inductive
limit topology.
In this section we focus on Radon measures (which by definition
are pos-itive but possibly unbounded) and positive linear
functionals on Cc(R). Forthese results it is the inductive limit
topology onCc(R) that will be important.In contrast, in Section
12.10 we will consider complex Radon measures (whichare necessarily
bounded) and corresponding linear functionals on Cc(R), andthere it
will be the L-topology on Cc(R) that will be important.
12.8.2 Positive Linear Functionals on Cc(R)
The next exercise shows that every Radon measure, bounded or
unbounded,induces a linear functional on Cc(R) that is continuous
with respect to theinductive limit topology on Cc(R). Further, this
functional is positive in thefollowing sense.
Definition 12.8.4. A functional : Cc(R) C is positive if f, 0
forall f Cc(R) with f 0.
Exercise 12.8.5. Let be a Radon measure on R. Define : Cc(R) C
by
f, =
f d, f Cc(R).
(a) Show that is a positive linear functional on Cc(R).
(b) Show that |C(K) : C(K) R is continuous for every compact
setK R,i.e.,
compact K R, CK > 0 such that
f C(K) = |f, | CK f. (12.11)
-
12.8 The Riesz Representation Theorem for Positive Functionals
on Cc(R) 451
The preceding exercise shows that those positive linear
functionals onCc(R) that are induced from Radon measures are
continuous with respectto the inductive limit topology on Cc(R).
Next we will show that every pos-itive linear functional on Cc(R)
is continuous with respect to the inductivelimit topology on
Cc(R).
Theorem 12.8.6. If : Cc(R) C is a positive linear functional on
Cc(R),then is continuous on Cc(R) with respect to the inductive
limit topology.That is, |C(K) : C(K) C is continuous for each
compact set K R.
Proof. Given a compact set K, Urysohns Lemma (Theorem 2.9.2)
impliesthat there exists a function K Cc(R) such that K 0 and K = 1
on K.
Suppose that f C(K) is real-valued. Then
|f(x)| = |f(x)| K(x) f K(x), x R.
Hence f K f 0, so
0 f K f,
= f K , f, .
Consequently,|f, | K , f.
Now let f C(K) be arbitrary. Then
|f, | |Re(f), |+ |Im(f), | 2 K , f,
so the result follows with CK = 2 K , .
Although we will not prove it, the Riesz Representation Theorem
com-pletes the characterization of positive linear functionals on
Cc(R): Everypositive linear functional on Cc(R) is induced from a
Radon measure.
Theorem 12.8.7 (Riesz Representation Theorem I). If : Cc(R) C is
a positive linear functional, then there exists a unique positive
Radonmeasure on R such that
f, =
f d, f Cc(R).
Moreover, if U R is open, then
(U) = sup{f, : f Cc(R), 0 f 1, supp(f) U
},
and if K R is compact then
(K) = inf{f, : f Cc(R), f K
}.
Thus, Radon measures and positive linear functionals on Cc(R)
are equiva-lent. Therefore, we often use the same symbol to
represent a Radon measure and the positive functional f 7 f, =
f d that it induces.
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452 12 Borel and Radon Measures on the Real Line
Problems
12.8.8. This problem will show that the locally finite positive
measures onN (which by Problem 12.7.5 are precisely the Radon
measures on N) are in1-1 correspondence with the positive linear
functionals on c00.
(a) Give the convergence criterion corresponding to the
inductive limittopology on c00.
(b) Show that if is a positive locally finite measure on N, then
f, =f(k) {k} defines a positive linear functional on c00 that is
continuous with
respect to the inductive limit topology on c00.
(c) Show that if is a positive linear functional on c00, then
there existsa unique sequence of nonnegative scalars w = (wk)kN
such that f, =
f(k)wk for f c00. Show there is a unique locally finite positive
measure on N such that wk = {k} for every k.
12.9 The Relation Between Radon and Borel Measures
We will use the Riesz Representation Theorem to show that every
locallyfinite positive Borel measure on R is a Radon measure (the
converse holdsby definition). First we need a lemma.
Lemma 12.9.1. If is a -finite Radon measure and E B, then for
every > 0 there exists an open set U and a closed set F such
that
F E U and (U\F ) < .
Proof. Since is -finite, there exist disjoint sets Ek B with
(Ek)
-
12.9 The Relation Between Radon and Borel Measures 453
Proof. By definition, if is a Radon measure, then it is a
locally finite positiveBorel measure.
Conversely, suppose that is a locally finite positive Borel
measure.We will show that is regular, and hence is a Radon measure.
SinceCc(R) L
1(), we can define f, =f d for f Cc(R), and this defines
a positive linear functional on Cc(R). The Riesz Representation
Theorem(Theorem 12.8.7) therefore implies that there exists a Radon
measure suchthat f, = f, for f Cc(R).
Now let U be any open subset of R. Then we can write U =j=1
Kj
where each Kj is compact. We claim that there exist functions fn
Cc(R)with 0 fn 1 and supp(fn) U such that fn = 1 on
nj=1 Kj and onn1
j=1 supp(fn).
To prove this, we proceed by induction. For n = 1, Urysohns
Lemma(Theorem 2.9.2) implies that there exists a function f1 Cc(R)
that satisfies0 f1 1, supp(f1) U, and f1 = 1 on K1.
Assume that f1, . . . , fn have been constructed satisfying the
required prop-erties. Then since
F =
(n+1j=1
Kj
) ( nj=1
supp(fj)
)is a compact subset of U, by Urysohns Lemma we can find a
function fn+1 Cc(R) such that 0 fn+1 1, supp(fn+1) U, and fn+1 = 1
on F. Thiscompletes the induction.
By construction, the sequence {fn}nN is monotone increasing and
fnconverges pointwise to U as n . Applying the Monotone
ConvergenceTheorem to both and , we see that
(U) =
U d = lim
n
fn d = lim
n
fn d =
U d = (U).
Thus and agree on all the open sets.Now let E be any Borel set,
and choose > 0. By Lemma 12.9.1, there exist
an open set U and a closed set F such that F E U and (U\F ) <
.Since U\F is open, and assign it the same measure, so (U\F ) <
.Therefore
(U) = (U\F ) + (F ) + (E).
Thus (E) = inf{(U) : U E, U open
}, so is outer regular on every
Borel set.Additionally,
(E) = (F ) + (E\F ) (F ) + .
Although F need not be compact, if we define Fk = F [k, k] then
Fkis compact and (Fk) (F ). If (E) < , then there exists a k
such
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454 12 Borel and Radon Measures on the Real Line
that (Fk) (F ) , and hence (Fk) (E) 2. If (E) = , then(F ) = as
well, and so (Fk) . In either case, we conclude that(E) = sup
{(K) : K E, K compact
}, so is inner regular on every
Borel set.Thus is regular, and hence is a Radon measure. In
fact, by the uniqueness
statement in the Riesz Representation Theorem, we actually have
= .
Corollary 12.9.3. The following statements are equivalent.
(a) is a locally finite positive Borel measure on R.
(b) is a regular locally finite positive Borel measure on R.
(c) is a Radon measure on R.
The following statements are also equivalent.
(a) is a bounded positive Borel measure on R.
(b) is a regular bounded positive Borel measure on R.
(c) is a bounded Radon measure on R.
More general domains on which the class of complex Borel
measures coin-cides with the class of complex Radon measures are
discussed in [Fol99].
Problems
12.9.4. Show that if is a Radon measure and f L1() with f 0,
thenf d is a Radon measure.
12.10 The Dual of C0(R)
In this section we will see that the dual space of C0(R) can be
identified withthe space of complex Radon measures on the real
line.
Radon measures, as discussed so far, are positive by definition.
We extendthe definition to signed and complex measures in the
expected manner.
Definition 12.10.1. A signed Borel measure on R is a signed
Radon mea-sure on R if its positive and negative parts +, are Radon
measures.
A complex Borel measure on R is a complex Radon measure on R if
itsreal and imaginary parts r, i are signed Radon measures.
Because of the properties of the real line, these notions
simplify as follows.
Lemma 12.10.2. The following statements are equivalent.
(a) is a bounded signed Borel measure on R.
-
12.10 The Dual of C0(R) 455
(b) is a bounded signed Radon measure on R.
The following statements are also equivalent.
(a) is a complex Borel measure on R.
(b) is a complex Radon measure on R.
Proof. A measure is a bounded signed Borel measure if and only
if +, are bounded positive Borel measures. By Theorem 12.9.2, this
happensif and only if +, are bounded Radon measures, which is
equivalent to being a bounded signed Radon measure.
A similar argument applies to complex measures, noting that all
complexmeasures are bounded by Exercise 12.5.2.
Consequently, the Banach spaceMb(R) of all complex Borel
measures on Rintroduced in Definition 12.5.14 coincides with the
space of all complex Radonmeasures on R. For domains other than R,
the distinction between these twospaces can be important.
The next exercise shows that if is a complex Radon measure
(thereforebounded), then induces a linear functional on Cc(R) that
is continuouswith respect to the uniform topology. Hence this
linear functional extends toa continuous linear functional on
C0(R).
Exercise 12.10.3. Assume that is a complex Radon measure, and
let be the complex conjugate measure defined in Problem 12.5.23.
Define a func-tional on Cc(R) by
f, =
f d, f Cc(R). (12.12)
In equation (12.12), we have used the complex conjugate measure
in orderto make the form f, antilinear in . Prove the following
statements.
(a) is bounded on Cc(R) with respect to the L-norm, and
|f, | f , f Cc(R), (12.13)
where = ||(R) is the norm of the measure .
(b) The operator norm of is = ||(R) = .
(c) extends to a bounded linear functional on C0(R), and this
functional isdefined by the rule f, =
f d for f C0(R).
To motivate the Riesz Representation Theorem for complex
measures on R,recall Exercise 9.5.11, which shows that c0
= 1, and Problem 12.5.26, whichshows that Mb(N) =
1. Combining those two problems we see that, in thediscrete
setting, the dual of c0 is the space of complex Radon measures on
N:
c0 = Mb(N).
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456 12 Borel and Radon Measures on the Real Line
Although we will not prove it, the Riesz Representation Theorem
states thatan analogous characterization holds on the real
line.
Theorem 12.10.4 (Riesz Representation Theorem II). Given a
mea-sure Mb(R), define : C0(R) C by
f, =
f d, f C0(R).
Then T : 7 is an antilinear isometry of Mb(R) onto C0(R).
Thus, C0(R) =Mb(R). We often write C0(R)
=Mb(R), meaning equal-ity in the sense of the identification
given in Theorem 12.10.4. Since Cc(R)is dense in C0(R) with respect
to the uniform topology, this implies thatCc(R)
= Mb(R). A complementary development of complex Radon mea-sures
could have started by declaring a complex Radon measure to be
anelement of the dual space of Cc(R) or C0(R) with respect to the
uniformtopology. We could go further in this direction and declare
the space of un-bounded complex Radon measures to be the dual space
of Cc(R) with respectto the inductive limit topology. Indeed,
Theorem 12.8.7 shows that the posi-tive Radon measures correspond
exactly to the positive linear functionals onCc(R) that are
continuous with respect to the inductive limit topology.
Problems
12.10.5. Show directly that if is an unbounded Radon measure on
R, thenthere exist functions fn Cc(R) with fn 0 and fn 1 such
thatfn, as n.
12.10.6. Given fn, f C0(R), show that fnw f if and only if sup
fn