On compactness of measures on Polish spaces

Abstract

We present some results related to the question whether every finite measure $\mu$ defined on a $\sigma$--algebra $\Sigma\sub \Borel[0,1]$ is countably compact. In particular, we show that for every finite measure space $(X,\Sigma,\mu)$, where $X$ is a Polish space and $\Sigma\sub \Borel(X)$, there is a regularly monocompact measure space $(\widehat{X},\widehat{\Sigma},\widehat{\mu})$ and an inverse-measure-preserving function $f:\widehat{X}\to X$.