Borel and Radon Measures

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

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 .

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.


If |(K)|


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


+(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)|.


Exercise 12.2.14. Let be a signed Borel measure. Prove that

is bounded ||(R)


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)

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)


(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

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.


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.

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.


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

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


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.

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)|


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

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).


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.


(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)|.


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.


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.


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


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).


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

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Borel and Radon Measures