On lifting invariant probability measures

Abstract

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