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Real and Complex Measures Chapter 9 Real and Complex Measures A measure is a countably additive function from a s-algebra to [0, ¥]. In this chapter, we consider countably additive functions from a s-algebra to either R or C. The first section of this chapter shows that these functions, called real measures or complex measures, form an interesting Banach space with an appropriate norm. The second section of this chapter focuses on decomposition theorems that help us understand real and complex measures. These results will lead to a proof that the dual space of Lp(m) can be identified with Lp0(m). Dome in the main building of the University of Vienna, where Johann Radon (1887–1956) was a student and then later a faculty member. The Radon–Nikodym Theorem, which will be proved in this chapter using Hilbert space techniques, provides information analogous to differentiation for measures. CC-BY-SA Hubertl © Sheldon Axler 2020 S. Axler, Measure, Integration & Real Analysis, Graduate Texts 255 in Mathematics 282, https://doi.org/10.1007/978-3-030-33143-6_9 256 Chapter 9 Real and Complex Measures 9A Total Variation Properties of Real and Complex Measures Recall that a measurable space is a pair (X, ), where is a s-algebra on X. Recall S S also that a measure on (X, ) is a countably additive function from to [0, ¥] that takes Æ to 0. Countably additiveS functions that take values in R orSC give us new objects called real measures or complex measures. 9.1 Definition real and complex measures Suppose (X, ) is measurable space. S A function n : F is called countably additive if • S! ¥ ¥ n Ek = ∑ n(Ek) = k[=1 k 1 for every disjoint sequence E , E ,... of sets in . 1 2 S A real measure on (X, ) is a countably additive function n : R. • S S! A complex measure on (X, ) is a countably additive function n : C. • S S! The word measure can be ambiguous The terminology nonnegative in the mathematical literature. The most measure would be more appropriate common use of the word measure is as than positive measure because the we defined it in Chapter 2 (see 2.54). function m : F defined by However, some mathematicians use the S! m(E) = 0 for every E is a word measure to include what are here positive measure. However,2 S we will called real and complex measures; they stick with tradition and use the then use the phrase positive measure to phrase positive measure. refer to what we defined as a measure in 2.54. To help relieve this ambiguity, in this chapter we usually use the phrase (positive) measure to refer to measures as defined in 2.54. Putting positive in paren- theses helps reinforce the idea that it is optional while distinguishing such measures from real and complex measures. 9.2 Example real and complex measures Let l denote Lebesgue measure on [ 1, 1]. Define n on the Borel subsets of • [ 1, 1] by − − n(E) = l E [0, 1] l E [ 1, 0) . \ − \ − Then n is a real measure. If m1 and m2 are finite (positive) measures, then m1 m2 is a real measure and • a m + a m is a complex measure for all a , a C−. 1 1 2 2 1 2 2 If n is a complex measure, then Re n and Im n are real measures. • Section 9A Total Variation 257 Note that every real measure is a complex measure. Note also that by definition, ¥ is not an allowable value for a real or complex measure. Thus a (positive) measure m on (X, ) is a real measure if and only if m(X) < ¥. Some authorsS use the terminology signed measure instead of real measure; some authors allow a real measure to take on the value ¥ or ¥ (but not both, because the expression ¥ ¥ must be avoided). However, real measures− as defined here serve us better because− we need to avoid ¥ when considering the Banach space of real or complex measures on a measurable space (see 9.18). For (positive) measures, we had to make m(Æ) = 0 part of the definition to avoid the function m that assigns ¥ to all sets, including the empty set. But ¥ is not an allowable value for real or complex measures. Thus n(Æ) = 0 is a consequence of our definition rather than part of the definition, as shown in the next result. 9.3 absolute convergence for a disjoint union Suppose n is a complex measure on a measurable space (X, ). Then S (a) n(Æ) = 0; ¥ ¥ (b) ∑ n(Ek) < for every disjoint sequence E1, E2,... of sets in . k=1j j S Proof To prove (a), note that Æ, Æ,... is a disjoint sequence of sets in whose union equals Æ. Thus S ¥ n(Æ) = ∑ n(Æ). k=1 The right side of the equation above makes sense as an element of R or C only when n(Æ) = 0, which proves (a). To prove (b), suppose E1, E2,... is a disjoint sequence of sets in . First suppose n is a real measure. Thus S n E = ∑ n(E ) = ∑ n(E ) k k j k j k : n(E )>0 k : n(Ek)>0 k : n(Ek)>0 f [k g f g f g and n E = ∑ n(E ) = ∑ n(E ) . − k − k j k j k : n(E )<0 k : n(Ek)<0 k : n(Ek)<0 f [k g f g f g Because n(E) R for every E , the right side of the last two displayed equations ∑2¥ 2¥ S is finite. Thus k=1 n(Ek) < , as desired. Now consider thej case wherej n is a complex measure. Then ¥ ¥ ¥ ∑ n(Ek) ∑ (Re n)(Ek) + (Im n)(Ek) < , k=1j j ≤ k=1 j j j j where the last inequality follows from applying the result for real measures to the real measures Re n and Im n. 258 Chapter 9 Real and Complex Measures The next definition provides an important class of examples of real and complex measures. 9.4 measure determined by an 1-function L Suppose m is a (positive) measure on a measurable space (X, ) and h 1(m). Define n : F by S 2 L S! n(E) = h dm. ZE Then n is a real measure on (X, ) if F = R and is a complex measure on (X, ) if F = C. S S Proof Suppose E , E , ... is a disjoint sequence of sets in . Then 1 2 S ¥ ¥ ¥ ¥ 9.5 n E = ∑ c (x)h(x) dm(x) = ∑ c h dm = ∑ n(E ), k Ek Ek k = = = k[=1 Z k 1 k 1 Z k 1 where the first equality holds because the sets E1, E2,... are disjoint and the second equality follows from the inequality m ∑ c (x)h(x) h(x) , Ek ≤ j j k=1 which along with the assumption that h 1( m) allows us to interchange the integral and limit of the partial sums by the Dominated2 L Convergence Theorem (3.31). The countable additivity shown in 9.5 means n is a real or complex measure. The next definition simply gives a notation for the measure defined in the previous result. In the notation that we are about to define, the symbol d has no separate meaning—it functions to separate h and m. 9.6 Definition h dm Suppose m is a (positive) measure on a measurable space (X, ) and h 1(m). S 2 L Then h dm is the real or complex measure on (X, ) defined by S (h dm)(E) = h dm. ZE Note that if a function h 1(m) takes values in [0, ¥), then h dm is a finite (positive) measure. 2 L The next result shows some basic properties of complex measures. No proofs are given because the proofs are the same as the proofs of the corresponding results for (positive) measures. Specifically, see the proofs of 2.57, 2.61, 2.59, and 2.60. Because complex measures cannot take on the value ¥, we do not need to worry about hypotheses of finite measure that are required of the (positive) measure versions of all but part (c). Section 9A Total Variation 259 9.7 properties of complex measures Suppose n is a complex measure on a measurable space (X, ). Then S (a) n(E D) = n(E) n(D) for all D, E with D E; n − 2 S ⊂ (b) n(D E) = n(D) + n(E) n(D E) for all D, E ; [ − \ 2 S ¥ (c) n Ek = lim n(Ek) k ¥ k[=1 ! for all increasing sequences E E of sets in ; 1 ⊂ 2 ⊂ · · · S ¥ (d) n Ek = lim n(Ek) k ¥ k\=1 ! for all decreasing sequences E E of sets in . 1 ⊃ 2 ⊃ · · · S Total Variation Measure We use the terminology total variation measure below even though we have not yet shown that the object being defined is a measure. Soon we will justify this terminology (see 9.11). 9.8 Definition total variation measure Suppose n is a complex measure on a measurable space (X, ). The total variation measure is the function n : [0, ¥] defined by S j j S! n (E) = sup n(E ) + + n(E ) : n Z+ and E ,..., E j j j 1 j ··· j n j 2 1 n are disjoint sets in such that E E E . S 1 [···[ n ⊂ To start getting familiar with the definition above, you should verify that if n is a complex measure on (X, ) and E , then S 2 S n(E) n (E); • j j ≤ j j n (E) = n(E) if n is a finite (positive) measure; • j j n (E) = 0 if and only if n(A) = 0 for every A such that A E.
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