Suppose $E$ is an ordered locally convex space, $X_1$ and $X_2$ Hausdorff completely regular spaces and $Q$ a uniformly bounded, convex and closed subset of $M_t^{+}(X_1\times X_2,E)$. For $i=1,2$, let ${\mu }_i\in M_t^{+}(X_i,E)$. Then, under some topological and order conditions on $E$, necessary and sufficient conditions are established for the existence of an element in $Q$, having marginals ${\mu }_1$ and ${\mu }_2$.

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 $X$ be a locally convex space, $m:\Sigma \to X$ be a vector measure defined on a $\sigma$-algebra $\Sigma$, and $L^1\left(m\right)$ be the associated (locally convex) space of $m$-integrable functions. Let $\Sigma \left(m\right)$ denote $\{{\chi }_{{}_E};E\in \Sigma \}$, equipped with the relative topology from $L^1\left(m\right)$. For a subalgebra $𝒜\subseteq \Sigma$, let ${𝒜}_{\sigma }$ denote the generated $\sigma$-algebra and ${\overline{𝒜}}_s$ denote the closure of $\chi \left(𝒜\right)=\{{\chi }_{{}_E};E\in 𝒜\}$ in $L^1\left(m\right)$. Sets of the form ${\overline{𝒜}}_s$ arise in criteria determining separability of $L^1\left(m\right)$; see [6]. We consider some natural questions concerning ${\overline{𝒜}}_s$ and, in particular, its relation to $\chi \left({𝒜}_{\sigma }\right)$. It is shown that...

It is proved that if X is infinite-dimensional, then there exists an infinite-dimensional space of X-valued measures which have infinite variation on sets of positive Lebesgue measure. In term of spaceability, it is also shown that $ca(ℬ,\lambda ,X)\setminus M_{\sigma }$, the measures with non-σ-finite variation, contains a closed subspace. Other considerations concern the space of vector measures whose range is neither closed nor convex. All of those results extend in some sense theorems of Muñoz Fernández et al. [Linear...

Let $(X,\parallel ·{\parallel }_X)$ be a real Banach space and let $E$ be an ideal of $L^0$ over a $\sigma$-finite measure space $(Ø,\Sigma ,\mu )$. Let $\left(X\right)$ be the space of all strongly $\Sigma$-measurable functions $fØ\to X$ such that the scalar function $\tilde{f}$, defined by $\tilde{f}\left(ø\right)={\parallel f\left(ø\right)\parallel }_X$ for $ø\in Ø$, belongs to $E$. The paper deals with strong topologies on $E\left(X\right)$. In particular, the strong topology $\beta (E\left(X\right),E{\left(X\right)}_n^{\sim })$ ($E{\left(X\right)}_n^{\sim }=$ the order continuous dual of $E\left(X\right)$) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.