On lifting invariant probability measures

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 $Y$.