On images of Borel measures under Borel mappings

Abstract:

Let $X$ and $Y$ be metric spaces. We show that the tight images of a (fixed) tight Borel probability measure $\mu$ on $X$, under all Borel mappings $f\colon X\to Y$, form a closed set in the space of tight Borel probability measures on $Y$ with the weak$^*$-topology. In contrast, the set of images of $\mu$ under all continuous mappings from $X$ to $Y$ may not be closed. We also characterize completely the set of tight images of $\mu$ under Borel mappings. For example, if $\mu$ is non-atomic, then all tight Borel probability measures on $Y$ can be obtained as images of $\mu$, and as a matter of fact, one can always choose the corresponding Borel mapping to be of Baire class 2.