Let $ℳ$ be the Banach space of real measures on a $\sigma$-ring $\mathbf{R}$, let ${ℳ}^{\text{'}}$ be its dual, let $E$ be a quasi-complete locally convex space, let $E^{\text{'}}$ be its dual, and let $\mu$ be an $E$-valued measure on $\mathbf{R}$. If is shown that for any $\theta \in {ℳ}^{\text{'}}$ there exists an element $\int \theta d\mu$ of $E$ such that $⟨x^{\text{'}}\circ \mu ,\theta ⟩=⟨\int \theta d\mu ,x^{\text{'}}⟩$ for any $x^{\text{'}}\in E^{\text{'}}$ and that the map$$\theta \to \int \theta d\mu :{ℳ}^{\text{'}}\to E$$is order continuous. It follows that the closed convex hull of $\mu \left(\mathbf{R}\right)$ is weakly compact.

Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). For a real Banach space $(X,|\left|·\right|{|}_X)$ let E(X) be a subspace of the space L⁰(X) of μ-equivalence classes of strongly Σ-measurable functions f: Ω → X and consisting of all those f ∈ L⁰(X) for which the scalar function ${\left|\right|f\left(·\right)\left|\right|}_X$ belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let $D_u(={f\in E\left(X\right):\left|\right|f\left(·\right)\left|\right|}_X\le u)$ stand for the order interval in E(X). For a real Banach space $(Y,|\left|·\right|{|}_Y)$ a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈ E⁺ the set...