More on the locally convex space $(M(X),\beta(X))$ of a locally compact Hausdorff space $X$

Abstract

In the previous paper [12] we introduced the definition of the strict topology $\beta(X)$ on the measure space $M(X)$ for a locally compact Hausdorff space $X$. In this paper, we consider on $M(X)$ the topology $\beta(X)$ and we show that $\beta(X)$ is the weak topology under all left multipliers induced by a function space on $M(X)$. We then show that $\beta(X)$ can be considered as a mixed topology. This result is not only of interest in its own right, but also it paves the way to prove that $(M(X),\beta(X))$ is a Mazur space and the locally convex space $(M(S),\beta(S))$, equipped with the convolution multiplication is a complete semitopological algebra, for a wide class of locally compact semigroups $S$.