On lifting invariant probability measures

TL;DR

This paper investigates conditions under which an invariant probability measure on a Borel space can be lifted to an invariant measure on a larger space via a surjective map, establishing a key compactness criterion.

Contribution

It proves that if the fibers of the surjection are compact, then the invariant measure lifts to an invariant measure on the extended space.

Findings

Invariant measure lifts when fibers are compact.

Provides a criterion for measure lifting in Borel spaces.

Connects measure invariance with fiber compactness.

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 $T \colon X \to X$ preserving $\mu$, a compact metric 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 fibers of $p$ are compact then $\mu$ lifts to an $S$-invariant measure on $Y$.