Countable additivity of Henstock-Dunford Integral and Orlicz Space

Abstract

Given a real Banach space $\mathcal{X}$ and probability space $(\Omega ,\Sigma ,\mu )$ we characterize the countable additivity of Henstock-Dunford integral for Henstock integrable function taking values in $X$ as those weakly measurable function $g:\Omega \to \mathcal{X}$ for which $\{y^{\ast }g:y^{\ast }\in B_{\mathcal{X}}^{\ast }\}$ is relatively weakly compact in some separable Orlicz space ${\mathcal{L}}^{\overline{\varphi }}(\mu ).$ We find relatively weakly compact in some Orlicz space with Henstock-Gel'fand integral.

13 pages