An ergodic theorem for non-invariant measures

Abstract

Given a space $X$, a $\sigma$-algebra $\mathfrak{B}$ on $X$ and a measurable map $T:X\to X$, we say that a measure $\mu$ is half-invariant if, for any $B\in \mathfrak{B}$, we have $\mu (T^{-1}(B)\le \mu (B)$. In this note we present a generalization of Birkhoff's Ergodic theorem to $\sigma$-finite half-invariant measures.

16 pages