Countable additivity of Henstock–Dunford integrable functions and Orlicz Space

Abstract

Suppose $${\mathcal {X}}$$ is a real Banach space and $$(\varOmega , \varSigma , \mu )$$ is a probability space. We characterize the countable additivity of Henstock–Dunford integrable functions taking values in $${\mathcal {X}}$$ as those weakly measurable function $$ g: \varOmega \rightarrow {\mathcal {X}}, $$ for which $$\{y^*g: y^* \in B_{\mathcal {X}}^* \} $$ is relatively weakly compact in some separable Henstock–Orlicz space (in briefly H–Orlicz space) $$ H^{\theta }(\mu )$$ , where $$B_{\mathcal {X}}^*$$ is the closed unit ball in $${\mathcal {X}}^{*}.$$ We find relatively weakly compactness of some H–Orlicz space of Henstock–Gel’fand integrable functions.