bvca(Σ, X) REVISITED

Abstract

Abstract. Assuming that ( , ) is a measurable space and X is a Banach space we provide a quite general sufficient condition on X for bvca( , X) (the Banach space of all X-valued countably additive measures of bounded variation equipped with the variation norm) to contain a copy of c 0 if and only if X does. Some well-known results on this topic are straightforward consequences of our main theorem.2010 Mathematics Subject Classification. 28B05, 46B03.1. Preliminaries. Throughout this paper X will be a Banach space over the field ‫ދ‬ of real or complex numbers. Our notation is standard [3,4]. If ( , ) is a measurable space, ca( , X) denotes the Banach space over ‫ދ‬ of all X-valued countably additive measures F on provided with the semivariation norm F and bvca( , X) stand for the Banach space of all X-valued countably additive measures F of bounded variation on equipped with the variation norm |F|. We represent by ca + ( ) the set of all positive and finite measures defined on . If ( , , μ) is a finite measure space, recall that a weakly μ-measurable function f : → X is said to be Dunford integrable if x * f ∈ L 1 (μ) for every x * ∈ X * . If f is Dunford integrable and E ∈ the map x * → E x * f dμ, denoted by (D) E f dμ, is a continuous linear form on X * . If (D) E f dμ ∈ X for each E ∈ then f is called Pettis integrable and one writes (P) E f dμ instead of (D) E f dμ. A strongly μ-measurable function f : → X is said to be Bochner integrable if f (ω) dμ(ω) < ∞. As usual we denote by L 1 (μ, X) the Banach space of all (equivalence classes of) μ-Bochner integrable functions equipped with the normRecall that a series ∞ n=1 x n in X is said to be weakly unconditionally Cauchy (wuC) if+ ( ) is purely atomic, then ca( , X) contains a copy of c 0 or ∞ if and only if X contains, respectively, a copy of c 0 or ∞ [5]. Assuming that X has the Radon-Nikodym property with respect to each μ ∈ ca + ( ), then bvca( , X) contains a copy of c 0 or ∞ if and only if X does [7]. As a consequence, if each μ ∈ ca + ( ) is purely atomic then bvca( , X) contains a copy of c 0 or ∞ if and only if X contains, respectively, a copy of c 0 or ∞ . If there exists a nonzero atomless measure μ ∈ ca + ( ), the latter statement is no longer true [11]. However, if the range space of the measures is a dual Banach space X * , then bvca( , X * ) contains a copy of c 0 or ∞ if and only if X * does [10].