 Solution

The Birkhoff ergodic theorem says that for a measure-preserving transformation on a finite measure space and $f\in L^1$, the averages $A_{n}f=n^{-1}{\sum }_{k=0}^{n-1}f\circ {\theta }^k$ converge almost everywhere to an invariant integrable function $f^{\ast }$. This limit is the conditional expectation of $f$ given the invariant sigma-algebra, and $\int f^{\ast }d\mu =\int fd\mu$. If the transformation is ergodic and $\mu (E)>0$, then $f^{\ast }$ equals the constant $\mu (E{)}^{-1}\int fd\mu$ almost everywhere.