On lifting invariant probability measures

Abstract

In this note we study when an invariant probability measure lifts to an invariant measure. Consider a standard Borel space $X$, a Borel probability measure $\mu$ on $X$, a Borel map $T:X\to X$ preserving $\mu$, a compact metric space $Y$, a continuous map $S:Y\to Y$, and a Borel surjection $p:Y\to X$ with $p\circ S=T\circ p$. We prove that if fibers of $p$ are compact then $\mu$ lifts to an $S$-invariant measure on $Y$.

Differs slightly from published version