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

Wolfgang Gr{ömig
1974 Proceedings of the American Mathematical Society  
It is known that a lot of topological properties devolve from a basic space X to the family MT(X) of all r-smooth Borel measures endowed with the weak topology (or to certain subspaces of Mr(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 d-compactness. For this purpose it is proved that
more » ... he countable product space XN is homeomorphic to a closed subset of MT(X). A further consequence of this result is for example that, for the family M](X) of probability measures in MT(X), compactness, local compactness and tr-compactness are equivalent properties. Let X be a Hausdorff space. By MiX) we denote the family of all nonnegative finite-valued measures defined on the Borel field of X. Let M(X) be endowed with the weak topology, i.e. the weakest topology for which each map p^-pG, where G is an open subset of X, is lower semicontinuous and p->pX is continuous. It is an immediate consequence of the definition of this topology that a net (pJaeD of measures converges (weakly) to a measure p in M(X) if and only if pX=lim pxX and either pG^lim inf pfi for every open set Ge X or pF^.lim sup pj? for every closed set F <= X. Further criteria for weak convergence of measures may be gathered from [3, Theorem 8.1]. Two subsets of M(X), the families MtiX) of tight and Mr(X) of rsmooth measures in M(X), are of special interest. A measure p e M(X) is said to be tight if pA = suo{pK | K <=■ A, K compact} for each Borel set A and r-smooth if pF0 = inf{pF\F e^}
doi:10.1090/s0002-9939-1974-0338758-7 fatcat:qtvzc56wj5h4fjfahqx2epjg5q