Weak Precompactness in the Space of Vector-Valued Measures of Bounded Variation

Let X be a Banach space, (Ω, Σ) a measure space, and M(Ω,X) the space of all X-valued countably additive measures on (Ω, Σ) of bounded variation (with the total variation norm). For m ∈ M(Ω,X) we denote by |m| its variation. For λ, a probability measure on (Ω, Σ), we denoteW(λ,X) = {m ∈ M(Ω,X) : |m| ≤ λ}. Ulger [1] and Diestel et al. [2] gave a characterization of weakly compact subsets of L(μ, X), the Banach space of all X-valued Bochner integrable functions on a probability space (Ω, Σ, μ). In [3] we gave a characterization of weakly precompact subsets of L(μ, X). Randrianantoanina and Saab [4] gave a characterization of relatively weakly compact subsets ofM(Ω,X). In this paper we use results of Talagrand [5], Ulger [6], and techniques of Randrianantoanina and Saab [4] to characterize weakly precompact subsets of M(Ω,X). The characterization is obtained in two steps. In the first step we characterize the weakly precompact subsets ofW(λ,X). We show that a subset A ofW(λ,X) is weakly precompact if and only if for any sequence (m n ) in A and for any lifting ρ of

In this paper we use results of Talagrand [5], Ulger [6], and techniques of Randrianantoanina and Saab [4] to characterize weakly precompact subsets of (Ω, ).The characterization is obtained in two steps.In the first step we characterize the weakly precompact subsets of (, ).We show that a subset  of (, ) is weakly precompact if and only if for any sequence (  ) in  and for any lifting  of  ∞ () there exists a sequence (   ) with    ∈ co{  :  ≥ } for each  such that, for a.e., the sequence ((   )()) is weakly Cauchy.In the second step we show that a subset  of (Ω, ) is weakly precompact if and only if there is a probability measure  on (Ω, Σ) such that, for any sequence (  ) in , there is a sequence (   ) with    ∈ co{  :  ≥ } for each  such that, for any  > 0, there is a positive integer  and a weakly precompact subset   of (, ) so that {   :  ≥ 1} ⊆   + (0), where (0) denotes the unit ball of (Ω, ).
This paper also contains several corollaries of these results.We show that if ℓ 1 ̸ → * , then a subset  of (Ω,  * ) is weakly precompact if and only if  is bounded and () = {|| :  ∈ } is uniformly countably additive.

Definitions and Notation
Throughout this paper,  and  will denote Banach spaces.The unit ball of  will be denoted by   .The unit basis of ℓ 1 will be denoted by ( *  ), and a continuous linear transformation  :  →  will be referred to as an operator.The set of all compact operators from  to  will be denoted by (, ).The set of all  * − continuous compact operators from  * to  will be denoted by   * ( * , ).
A bounded subset  of  is said to be weakly precompact provided that every sequence from  has a weakly Cauchy subsequence [5].For a subset  of , let co() denote the convex hull of .A series ∑   in  is said to be weakly unconditionally convergent (wuc) if for every  * ∈  * the series ∑ | * (  )| is convergent.An operator  :  →  is weakly precompact if (  ) is weakly precompact and unconditionally converging if it maps weakly unconditionally convergent series to unconditionally convergent ones.
Suppose Ω is a compact Hausdorff space,  and  are Banach spaces, (Ω, ) is the Banach space of all continuous -valued functions (with the supremum norm), and Σ is the -algebra of Borel subsets of Ω.It is known from [11] that (Ω, ) * ≃ (Ω,  * ).
The following results will be useful in our study.
For the proofs of the following two theorems we will need the following lemmas.The first lemma contains a well-known result due to Grothendieck about relatively weakly compact sets (see [18, Note that  is a well-defined operator, ∑  * (  ) is wuc, and ⟨  ,  * (  )⟩ = ∫     >  for each .Then {  :  ≥ 1} is not a  * -set.This contradiction concludes the proof.
Proof.(i) The proof is similar to the proof of Theorem 13(i).