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.

假设\（{\ mathcal {X}} \）是一个真实的Banach空间，而\（（（\ varOmega，\ varSigma，\ mu）\）是一个概率空间。我们表征Henstock可-邓福德积函数取值在可数加\（{\ mathcal {X}} \）的那些弱测函数\（克：\ varOmega \ RIGHTARROW {\ mathcal {X}}，\）为其\（\ {Y ^ *克为：y ^ * \在乙_ {\ mathcal {X}} ^ * \} \）是在一些可分离Henstock可-Orlicz空间相对弱紧（以简要H-Orlicz空间）\（H ^ {\ theta}（\ mu）\），其中\（B _ {\ mathcal {X}} ^ * \）是\（{\ mathcal {X}} ^ {*}。\\中的闭合单位球。 我们发现Henstock-Gel'fand可积函数的某些H-Orlicz空间的相对较弱的紧致性。