Compact measures and measurable weak sections

Abstract

Compact measures, i.e. measures that are inner-regular with respect to a compact family of sets, are related to measurable weak sections in the same way as semicompact measures are related to disintegration. This enables us to prove several stability properties of the class of compact measures. E.g., a countable sum of compact measures is compact; the image $\nu$ of a compact measure $\mu$ is compact provided $\mu$ is an extremal preimage measure of $\nu$. As a consequence, we show that every tight Baire measure is compact.