Sequences in the range of a vector measure and Banach spaces satisfying Π₁(X, l ₁) = Π₂(X,l ₁)

Abstract

If X is a Banach space such that Π₁(X,l ₁) = Π₂ (X, l ₁), we prove the following results: 1) A bounded sequence (x _n ) lies inside the range of some X-valued measure if and only if the operator (α _n ) ∈ l ₁ → Σ _n α _n x _n ∈ X is 1-summing, and 2) If A is a bounded subset of X lying in the range of some X ^**-valued measure, then A is necessarily contained in the range of some X-valued measure.