On Sums of Pettis Integrable Random Elements

Abstract

Given a complete probability space (Ω, σ, μ) and a Banach space X, we investigate the relationship among different types of convergence in P ₁ (μ, X), the space of all X-valued μ -measurable [classes of] Pettis integrable functions, of sums of random elements of the form σ_{i =1} ε_if_i(where ε _itakes the values ± 1 with the same probability and f _i∈ P₁(μ, X) for each i ∈ N) and the existence of weakly unconditionally Cauchy subseries of σ^∞ _{n =1} fn (ω) for μ-almost all ω ∈ Ω.