 Solution

The Birkhoff ergodic theorem states that if $T$ is a measure-preserving transformation and $f\in L^1(\mu )$, thenwhere $\mathcal{I}$ is the invariant sigma-algebra. The limit has the same integral as $f$. If $T$ is an ergodic transformation, the limit is the constant ${\int }_{X}fd\mu$ almost everywhere.