Countable additivity of Henstock-Dunford Integral and Orlicz Space

TL;DR

This paper characterizes when the Henstock-Dunford integral is countably additive for functions in Banach spaces, linking it to relative weak compactness in certain Orlicz spaces.

Contribution

It provides a new characterization of countable additivity of the Henstock-Dunford integral via weak compactness in Orlicz spaces.

Findings

Countable additivity characterized by weak compactness in Orlicz spaces.

Weakly measurable functions with certain properties are linked to Henstock-Dunford integrability.

Identification of conditions for relative weak compactness in Orlicz spaces.

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^*g~: y^* \in B_\mathcal{X}^* \} $ is relatively weakly compact in some separable Orlicz space $ \mathcal{L}^{\overline{\phi}}(\mu) .$ We find relatively weakly compact in some Orlicz space with Henstock-Gel'fand integral.