On lifting invariant probability measures

Tomasz Cieśla¹

¹ Department of Mathematics and Statistics McGill University 805, Sherbrooke Street West Montreal, Quebec, Canada H3A 2K6

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 68 (2020) no. 1, pp. 69-74.

Résumé

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 $T \colon X \to X$ preserving $\mu $, a Polish space $Y$, a continuous map $S\colon Y \to Y$, and a Borel surjection $p \colon Y \to X$ with $p\circ S = T \circ p$. We prove that if the fibers of $p$ are compact then $\mu $ lifts to an $S$-invariant measure on $Y$.

Détail

DOI : 10.4064/ba200225-9-3

Keywords: study invariant probability measure lifts invariant measure consider standard borel space borel probability measure borel map colon preserving polish space continuous map colon borel surjection colon circ circ prove fibers compact lifts s invariant measure