Absolute Continuity of Vitali–Hahn–Saks Measure Convergence Theorems

Abstract

In this paper, we prove the following improved Vitali–Hahn–Saks measure convergence theorem: Let (L, 0, 1) be a Boolean algebra with the sequential completeness property, (G, τ) be an Abelian topological group, ν be a nonnegative finitely additive measure defined on L, {μ_n: n∈ N} be a sequence of finitely additive s-bounded G-valued measures defined on L, too. If for each a∈ L, {μ_n(a)}_{n∈ N} is a τ-convergent sequence, for each n∈N, when {ν (a_α)}_α∈Λ convergent to 0, {μ_n(a_α)}_α∈Λ is τ-convergent, then when {ν (a_α)}_α∈Λ convergent to 0, {μ_n(a_α)}_α∈Λ are τ-convergent uniformly with respect to n∈N