Chapter 1 Vector Measures In this chapter we present a survey of known results about vector mea- sures. For the convenience of the readers some of the results are given with proofs, but neither results nor proofs pretend to be ours. The only exception is the material of Section 1.5 that covers the results of our joint paper with V. M. Kadets [18]. 1.1 Elementary properties The results of this section may be found in [7]. Definition 1.1.1 A function µ from a field F of subsets of a set Ω to a Banach space X is called a finitely additive vector measure, or simply a vector measure, if whenever A1 and A2 are disjoint members of F then µ (A1 A2)=µ(A1)+µ(A2). ∪ ∞ ∞ If in addition µ An = µ (An) in the norm topology of X n=1 n=1 for all sequences (AnS) of pairwiseP disjoint members of F such that ∞ An F ,thenµis termed a countably additive vector measure, or n=1 ∈ simplyS µ is countably additive. Example 1.1.2 Finitely additive and countably additive vector mea- sures. Let Σ be a σ-field of subsets of a set Ω, λ be a nonnegative countably additive measure on Σ. Consider 1 p . Define ≤ ≤∞ 3 4 CHAPTER 1. VECTOR MEASURES µp :Σ Lp(Ω, Σ,λ)bytheruleµp(A)=χA for each set A Σ → ∈ (χA denotes the characteristic function of A). It is easy to see that µp is a vector measure, which is countably additive in the case 1 p< and fails to be countably additive in the case p = . ≤ ∞ ∞ Definition 1.1.3 A vector measure µ : F X is said to be nonatomic → if for every set A F with µ (A) =0there exist A1,A2 F Ø such ∈ 6 ⊂ \ that A1 A2 = A and µ (Ai) =0(i=1,2). ∪ 6 Definition 1.1.4 Let µ : F X be a vector measure. The variation of µ is the nonnegative function→ µ whose value on a set A F is given by | | ∈ µ (A)=sup µ(B) , | | π B πk k X∈ where the supremum is taken over all partitions π of A into a finite number of pairwise disjoint members of F .If µ(Ω) < , then µ will be called a measure of bounded variation. | | ∞ The semivariation of µ is the extended nonnegative function µ whose value on a set A F is given by k k ∈ µ (A)=sup x∗µ (A):x∗ X∗, x∗ 1 , k k {| | ∈ k k≤ } where x∗µ is the variation of the real-valued measure x∗µ. | | Example 1.1.5 A measure of bounded variation. Let µ1 be the mea- sure discussed in Example 1.1.2. Since µ1 (A) λ(A), it is plain k k≤ that µ1 (A) λ (A), so that µ1 is of bounded variation. | | ≤ The next proposition presents two basic facts about the semivaria- tion of a vector measure. Proposition 1.1.6 Let µ : F X be a vector measure. Then for A F one has → ∈ (a) µ (A)= sup εnµ (Bn) , where the supremum is taken k k εn ,π Bn π k k over all partitions{ } Pπ∈ of A into a finite number of pairwise disjoint members of F and over all finite collections εn satisfying εn 1; { } | |≤ 1.1. ELEMENTARY PROPERTIES 5 (b)sup µ(B) : A B F µ (A) 2sup µ(B) : A B F {k. Consequentlyk ⊇ a∈ vector}≤k measurek is≤ of bounded{k semivariationk ⊇ ∈ on}Ω if and only if its range is bounded in X. It is easy to define the integral of a bounded measurable function with respect to a bounded vector measure. To this end, let F be a field of subsets of a set Ω and µ : F X be a bounded vector measure. → n If f is a scalar-valued simple function, say f = αiχAi ,whereαi i=1 are nonzero scalars and A1, ..., An are pairwise disjointP members of F , n define Tµ (f)= αiµ (Ai). It is easy to show that this formula defines i=1 a linear operatorPTµ from the space of simple functions of the above form into X.Moreover,iffis as above and β =sup f (ω) : ω Ω , then {| | ∈ } n n αi Tµ (f) = αiµ (Ai) = β µ (Ai) β µ (Ω) . k k β ≤ k k i=1 i=1 X X by Proposition 1.1.6 (a). Thus Tµ is a continuous linear operator, which acts on the space of simple functions on F , equipped with the supre- mum norm. Another look at the Proposition 1.1.6 (a)andtheabove calculations shows that Tµ = µ (Ω) (1.1) k k k k Then Tµ has a unique continuous linear extension, still denoted by Tµ, to B (µ), the space of all scalar-valued functions on Ω that are uniform limits of simple functions modeled on F . (Note that in the case F is a σ-field, B (µ) is precisely the familiar space of bounded F -measurable scalar-valued functions defined on Ω). This discussion allows us to make Definition 1.1.7 Let F be a field of subsets of a set Ω and µ : F X be a bounded vector measure. For each f B (µ), the integral fdµ→ is ∈ defined by R fdµ = Tµ (f), Z where Tµ is as above. 6 CHAPTER 1. VECTOR MEASURES It is easy to see that this integral is linear in f (and also in µ)and satisfies fdµ f µ (Ω) . Z ≤k k∞k k Moreover, if x∗ X∗ ,then x∗ fdµ = fdx∗µ; indeed, for simple ∈ functions this equality is trivial andR densityR of simple functions in B (µ) proves the identity for all f B (µ). ∈ Definition 1.1.8 A family µτ : F X τ T of countably addi- tive vector measures is said{ to be uniformly→ | countably∈ } additive when- ever for any sequence (An) of pairwise disjoint members of F , the series ∞ µτ (An) converges in norm uniformly in τ T . n=1 ∈ P There are many alternative and useful formulations of countable additivity. We present two of them. Proposition 1.1.9 Any one of the following statements about a vector measure µ defined on a field F implies all the others. (i) µ is countably additive. (ii)Theset x∗µ:x∗ X∗, x∗ 1 is uniformly countably additive. { ∈ k k≤ } (iii)Theset x∗µ : x∗ X∗, x∗ 1 is uniformly countably addi- tive. {| | ∈ k k≤ } 1.2 Bartle-Dunford-Schwartz' theorem The results of this section may be found in [7]. Definition 1.2.1 Let F be a field of subsets of a set Ω, µ : F X be a vector measure, and λ be a finite nonnegative real-valued measure→ on F .If lim µ (A)=0, then µ is called λ-continuous and this is λ(A) 0 denoted by µ → λ. Sometimes we will say µ is absolutely continuous with respect toλ. 1.2. BARTLE-DUNFORD-SCHWARTZ' THEOREM 7 It should be noted that writing µ λ is not the same as saying µ vanishes on λ-null sets unless both λ and µ are countably additive and defined on a σ-field. Theorem 1.2.2 (Pettis) Let Σ be a σ-field of subsets of a set Ω, µ :Σ Xbe a countably additive vector measure, and λ be a finite nonnegative→ real-valued measure on Σ. Then µ is λ-continuous if and only if µ vanishes on sets of λ-measure zero. Proof. To prove the sufficiency, suppose µ vanishes on sets of λ- measure zero, but lim µ (A) > 0. Then there exist an ε>0anda λ(A) 0 → sequence (An) in Σ, such that n µ (An) εand λ (An) 2− k k≥ ≤ for all n.Foreachnselect an x∗ X∗ such that n ∈ ε x∗ 1and x∗µ(An) . k nk≤ | n |≥2 As µ is countably additive on Σ, then in accordance with Proposition 1.1.9 the set x∗µ : x∗ X ∗, x∗ 1 is uniformly countably addi- {| | ∈ k k≤ } ∞ ∞ tive. Now set Bn = Ai.Evidentlyλ Bn = 0. Consequently µ i=n n=1 S T ∞ vanishes on every set D Σ that is contained in Bn = B. It follows ∈ n=1 that x∗µ (B)=0. T | | Let C1 =Ω B1 and Cn+1 = Bn Bn+1 for all n.Then(Cn) constitutes a sequence\ of pairwise disjoint\ members of Σ for which ∞ Bn 1 B = Ci. − \ i=n [ Also, since x∗ µ (B) = 0 for all n, one has | n | ∞ ∞ lim xn∗ µ (Bm) = lim xn∗ µ Ci = lim xn∗ µ (Ci)=0 m | | m | | ! m | | i=[m iX=m uniformly in n, by the uniform countable additivity of the family xn∗ µ . But now {| |} ε xn∗ 1µ (Bn 1) xn∗ 1µ (An 1) xn∗ 1µ (An 1) . − − ≥ − − ≥ − − ≥ 2 8 CHAPTER 1. VECTOR MEASURES This contradicts the last calculation and proves the sufficiency; the converse is transparent. The following theorem is central to the theory of vector measures. Theorem 1.2.3 (Bartle-Dunford-Schwartz)Letµbe a bounded countably additive vector measure defined on a σ-field Σ. Then there exists a nonnegative real-valued countably additive measure λ on Σ such that µ is absolutely continuous with respect to λ.Ifµis nonatomic, then λ can be chosen nonatomic. (λ can be chosen such that 0 λ(A) µ (A) for all A Σ.) ≤ ≤ k k ∈ Corollary 1.2.4 Let Σ be a σ-field, µ :Σ Xbe a bounded countably additive vector measure, λ be a measure→ from Theorem 1.2.3.