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It is proved that if a Frechet space $E$ has $R-N$ property, then $L_p(E,\nu )$ also has $R-N$ property, for $1\<p\<\infty$.

Let $X$ be a completely regular Hausdorff space, $C_b\left(X\right)$ the space of all scalar-valued bounded continuous functions on $X$ with strict topologies. We prove that these are locally convex topological algebras with jointly continuous multiplication. Also we find the necessary and sufficient conditions for these algebras to be locally $m$-convex.

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 ${\mu }_2$.

For $i=(1,2)$, let $X_i$ be completely regular Hausdorff spaces, $E_i$ quasi-complete locally convex spaces, $E=E_1\breve{\otimes }{E}_2$, the completion of the their injective tensor product, $C_b\left(X_i\right)$ the spaces of all bounded, scalar-valued continuous functions on $X_i$, and ${\mu }_i$ $E_i$-valued Baire measures on $X_i$. Under certain conditions we determine the existence of the $E$-valued product measure ${\mu }_1\otimes {\mu }_2$ and prove some properties of these measures.

For a Banach space $E$ and a probability space $(X,𝒜,\lambda )$, a new proof is given that a measure $\mu :𝒜\to E$, with $\mu \ll \lambda$, has RN derivative with respect to $\lambda$ iff there is a compact or a weakly compact $C\subset E$ such that ${\left|\mu \right|}_C:𝒜\to [0,\infty ]$ is a finite valued countably additive measure. Here we define ${\left|\mu \right|}_C\left(A\right)=sup\{{\sum }_k|\langle \mu \left(A_k\right),f_k\rangle \left|\right\}$ where $\left\{A_k\right\}$ is a finite disjoint collection of elements from $𝒜$, each contained in $A$, and $\left\{f_k\right\}\subset E^{\text{'}}$ satisfies ${sup}_k\left|f_k\left(C\right)\right|\le 1$. Then the result is extended to the case when $E$ is a Frechet space.

Let $X$ be a completely regular $T_1$ space, $E$ a boundedly complete vector lattice, $C\left(X\right)$ $\left(C_b\left(X\right)\right)$ the space of all (all, bounded), real-valued continuous functions on $X$. In order convergence, we consider $E$-valued, order-bounded, $\sigma$-additive, $\tau$-additive, and tight measures on X and prove some order-theoretic and topological properties of these measures. Also for an order-bounded, $E$-valued (for some special $E$) linear map on $C\left(X\right)$, a measure representation result is proved. In case $E_n^{*}$ separates the points...

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, which extend...