 Solution

Let $(X,\mathcal{B},\mu )$ be a probability space, let $T$ be a measure-preserving transformation, and let $g\in L^1(\mu )$. The Birkhoff ergodic theorem asserts thatfor almost every $x$, where $\mathcal{I}=\{B\in \mathcal{B}:\mu (T^{-1}B△B)=0\}$ is the invariant sigma-algebra. The limit is integrable, satisfies $g^{\ast }\circ T=g^{\ast }$ almost everywhere, and has $\int g^{\ast }d\mu =\int gd\mu$. On a probability space the convergence also holds in $L^1$. No invertibility of $T$ is required.

If $T$ is ergodic, its invariant sigma-algebra is trivial modulo null sets, so its conditional expectation is constant. In that case the concise form is