Integral Representations for Continuous Linear Operators in the Setting of Convex Topological Vector Spaces

Abstract

Suppose $X$ and $Y$ are locally convex Hausdorff spaces, $H$ is arbitrary and $\Sigma$ is a ring of subsets of $H$. The authors prove the analog of the theorem stated in [Abstract 672-372, Notices Amer. Math. Soc. 17 (1970), 188] in this setting. A theory of extended integration on function spaces with Lebesgue and non-Lebesgue type convex topologies is then developed. As applications, integral representations for continuous transformations into $Y$ for the following function spaces $F$ (which have domain $H$ and range $X$) are obtained: (1) $H$ and $\Sigma$ are arbitrary, $\tau$ is a convex topology on the simple functions over $\Sigma ,K$ is a set function on $\Sigma$ with values in $L[X,Y]$, and $F$ is the Lebesgue-type space generated by $K$; (2) $H$ is a normal space and $F$ is the space of continuous functions each of whose range is totally bounded, with the topology of uniform convergence; (3) $H$ is a locally compact Hausdorff space, $F$ is the space of continuous functions of compact support with the topology of uniform convergence; (4) $H$ is a locally compact Hausdorff space and $F$ is the space of continuous functions with the topology of uniform convergence on compact subsets. In the above $X$ and $Y$ may be replaced by topological Hausdorff spaces under certain additional compensating requirements.