Let $X$ be a completely regular Hausdorff space, $E$ a boundedly complete vector lattice, $C_b\left(X\right)$ the space of all, bounded, real-valued continuous functions on $X$, $ℱ$ the algebra generated by the zero-sets of $X$, and $\mu C_b\left(X\right)\to E$ a positive linear map. First we give a new proof that $\mu$ extends to a unique, finitely additive measure $\mu ℱ\to E^{+}$ such that $\nu$ is inner regular by zero-sets and outer regular by cozero sets. Then some order-convergence theorems about nets of $E^{+}$-valued finitely additive measures on $ℱ$ are proved,...

Let E be a locally convex topological Hausdorff space, K a nonempty compact convex subset of E, μ a regular Borel probability measure on E and γ > 0. We say that the measure μ γ-represents a point x ∈ K if $su{p}_{\left|\right|f\left|\right|\le 1}|f\left(x\right)-{\int }_{K}fd\mu |<\gamma$ for any f ∈ E*. In this paper a continuous version of the Choquet theorem is proved, namely, if P is a continuous multivalued mapping from a metric space T into the space of nonempty, bounded convex subsets of a Banach space X, then there exists a weak* continuous family $\left({\mu }_t\right)$ of...

A closed convex set $Q$ in a local convex topological Hausdorff spaces $X$ is called locally nonconical (LNC) if for every $x,y\in Q$ there exists an open neighbourhood $U$ of $x$ such that $(U\cap Q)+\frac{1}{2}(y-x)\subset Q$. A set $Q$ is local cylindric (LC) if for $x,y\in Q$, $x\ne y$, $z\in (x,y)$ there exists an open neighbourhood $U$ of $z$ such that $U\cap Q$ (equivalently: $\mathrm{b}d\left(Q\right)\cap U$) is a union of open segments parallel to $[x,y]$. In this paper we prove that these two notions are equivalent. The properties LNC and LC were investigated in [3], where the implication $\mathrm{L}NC\Rightarrow \mathrm{L}C$ was proved in...

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.