Über die dem Lebesgueschen Mass isomorphen topologischen Massräume

Abstract

Homomorphisms of topological measure spaces had been defined in [5] to be measure-preserving and almost everywhere continuous mappings; this induces a concept of isomorphic topological measures. The main result of the present paper is that a locally finite atomfree measure μ in a completely regular space X with a countable base is isomorphic to the Lebesgue measure λ in an interval (case of finite measure) or the real line R (case of infinite measure) if and only if X contains a Polish subspace P such that μ(X-P)=0. A corollary states that any measure satisfying these conditions is carried by a G_δ-subset of X which can be mapped onto a G_δ-subset of R by a μ-λ-measure-preserving homeomorphism. Measures with atomic components are also treated. Further theorems concern measures in non-metrizable spaces or spaces without a countable base. For example it is proved that all compactifications (in the sense of topological measures) of a tight topological measure space are isomorphic, and they are isomorphic to the space itself if this space admits a complete metric.