Conditional weak compactness in vector-valued function spaces

Let $E$ be an ideal of $L^{0}$ over a $\sigma$-finite measure space $(\Omega ,\Sigma ,\mu )$ and let $E^{\prime }$ be the Köthe dual of $E$ with $\hbox {supp}\,E^{\prime }=\Omega$. Let $(X,\Vert\cdot \Vert _{X})$ be a real Banach space, and $X^{*}$ the topological dual of $X$. Let $E(X)$ be a subspace of the space $L^{0}(X)$ of equivalence classes of strongly measurable functions $f\colon \, \Omega \to X$ and consisting of all those $f\in L^{0}(X)$ for which the scalar function $\Vert f(\cdot )\Vert _{X}$ belongs to $E$. For a subset $H$ of $E(X)$ for which the set $\{\Vert f(\cdot )\Vert _{X}\colon \, f\in H\}$ is $\sigma (E,E^{\prime })$-bounded the following statement is equivalent to conditional $\sigma (E(X),E^{\prime }(X^{*}))$-compactness: the set $\{\Vert f(\cdot )\Vert _{X}\colon \, f\in H\}$ is conditionally $\sigma (E,E^{\prime })$-compact and $\{\int _{A} f(\omega )d\mu \colon \, f\in H\}$ is a conditionally weakly compact subset of $X$ for each $A\in \Sigma$, $\mu(A)<\infty$ with $\chi _{A}\in E^{\prime }$. Applications to Orlicz-Bochner spaces are given.