Birkhoff pointwise ergodic theorem

Statement

For a probability-preserving transformation $T$ and $f\in L^1(X,\mu)$ , the Birkhoff pointwise ergodic theorem states

\frac1N\sum_{n=0}^{N-1}f(T^nx) \longrightarrow \mathbb E[f\mid\mathcal I_T](x) \quad\text{for almost every }x.

Here $\mathcal I_T$ is the invariant sigma-algebra. Invertibility is not required. If $T$ is ergodic, the limit is $\int_X f\,d\mu$ .

Visit frequencies

Taking $f=1_E$ in an ergodic system identifies the asymptotic fraction of visits to $E$ with $\mu(E)$ . The exceptional null set may depend on $E$ or $f$ ; the theorem does not give a single exceptional set valid for all measurable observables at once.

Why this differs from mean convergence

The mean ergodic theorem concerns Hilbert-space norm convergence. Birkhoff's theorem concerns individual trajectories. A proof uses the maximal ergodic inequality to control exceptional sets, not merely the $L^2$ projection argument.

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. §2.5.
Anders Karlsson, “A proof of the subadditive ergodic theorem”, §1, Birkhoff theorem for a measure-preserving transformation without an invertibility assumption.