An ergodic theorem for non-invariant measures

TL;DR

This paper generalizes Birkhoff's Ergodic theorem to include  measures that are half-invariant, broadening the scope of ergodic theory to more general measure types.

Contribution

It introduces a new ergodic theorem applicable to  measures that are only half-invariant, extending classical results.

Findings

Generalizes Birkhoff's theorem to half-invariant measures

Establishes ergodic properties for  measures

Broadens applicability of ergodic theory

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 $\mu(T^{-1}(B)\leq \mu (B)$. In this note we present a generalization of Birkhoff's Ergodic theorem to $\sigma$-finite half-invariant measures.