Applications of the theory of Orlicz spaces to vector measures (Q2416489)

Let $(\Omega,\Sigma,\lambda)$ be a finite complete measure space and $E$ a sequentially complete locally convex Hausdorff space. The main theorem gives various characterizations of uniform $\lambda$-absolute continuity for uniformly bounded subsets $\mathcal{M}$ of $\text{ca}_\lambda(\Sigma,E)$. In particular, $\mathcal{M}$ is uniformly $\lambda$-absolutely continuous iff, for every equicontinuous subset $D$ of the topological dual $E'$, there exists a submultiplicative Young function $\varphi$ such that the set $\{\frac{d(e'\circ m)}{d\lambda}: m\in\mathcal{M},\, e'\in D\}$ is relatively weakly compact in the Orlicz space $L^\varphi(\lambda)$.