On a weakly closed subset of the space of $\tau$-smooth measures

Abstract:

It is known that a lot of topological properties devolve from a basic space $X$ to the family ${M_\tau }(X)$ of all $\tau$-smooth Borel measures endowed with the weak topology (or to certain subspaces of ${M_\tau }(X)$). The aim of this paper is to show that among these topological properties there cannot be properties which are hereditary on closed subsets but not on countable products of $X$, e.g. normality, paracompactness, the Lindelöf property, local compactness and $\sigma$-compactness. For this purpose it is proved that the countable product space ${X^N}$ is homeomorphic to a closed subset of ${M_\tau }(X)$. A further consequence of this result is for example that, for the family $M_\tau ^1(X)$ of probability measures in ${M_\tau }(X)$, compactness, local compactness and $\sigma$-compactness are equivalent properties.