Absolute Continuity of Vitali–Hahn–Saks Measure Convergence Theorems

Author: Junde Wu   Su Zhou   Minhyung Cho  

Publisher: Springer Publishing Company

ISSN: 0020-7748

Source: International Journal of Theoretical Physics, Vol.43, Iss.6, 2004-06, pp. : 1433-1436

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

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, {μn: n∈ N} be a sequence of finitely additive s-bounded G-valued measures defined on L, too. If for each a∈ L, {μn(a)}n∈ N is a τ-convergent sequence, for each n∈N, when {ν (aα)}α∈Λ convergent to 0, {μn(aα)}α∈Λ is τ-convergent, then when {ν (aα)}α∈Λ convergent to 0, {μn(aα)}α∈Λ are τ-convergent uniformly with respect to n∈N