= 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 that
$$
A_ng(x)=\frac1n\sum_{k=0}^{n-1}g(T^kx)
\longrightarrow g^*(x)=\mathbb E_\mu[g\mid\mathcal I](x)
$$
for almost every $x$, where $\mathcal I=\{B\in\mathcal B:\mu(T^{-1}B\mathbin{\triangle}B)=0\}$ is the <invariant sigma-algebra>. The limit is integrable, satisfies $g^*\circ T=g^*$ almost everywhere, and has $\int g^*\,d\mu=\int g\,d\mu$. 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
$$
\boxed{A_ng(x)\longrightarrow\int_Xg\,d\mu\quad\text{for almost every }x.}
$$
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