Functions with values in locally convex spaces with weakly compact semivariation

Abstract

The present paper is concerned with some properties of functions
with values in locally convex vector spaces, especially functions having weakly compact semivariation and  generalizations of some  theorems for  functions with values in locally convex vector spaces, namely: If $X$ is a sequentially complete locally convex vector space, then the function $x(\cdot):[a,b] \to X$ having a weakly compact semivariation on the interval $[a,b]$ defines  a vector valued measure $m$ on Borel subsets of $[a,b]$ with values in $X$ and the range of this measure is a weakly relatively compact subset in $X$. This theorem is an extension of the result of Sirvint  and of Edwards from Banach spaces to locally convex spaces.