Birkhoff's ergodic theorem

Remarks

The usual proof combines the maximal ergodic theorem with a density argument. For functions of the form $h + g - g \circ T$, with $h$ invariant and $g$ bounded, the averages converge everywhere, because the sums of $g - g \circ T$ telescope. The maximal inequality bounds $\sup_n \abs{A_n f}$ in measure by $\norm{f}_1$, and this lets convergence pass to all integrable $f$. For $f \in L^2(\mu)$, convergence in $L^2$ is the content of the mean ergodic theorem.

Example: let $T(x) = 2x \bmod 1$ on $[0, 1)$ with Lebesgue measure, which is ergodic, and let $f$ be the indicator of $[1/2, 1)$. Then $f(T^k(x))$ is the $(k+1)$-st binary digit of $x$, so the theorem says that almost every $x$ has, in the limit, half of its binary digits equal to $1$.

Applied to a sequence of independent identically distributed random variables, viewed as a shift on a product space, the theorem gives the strong law of large numbers.

The measure must be invariant: for $T(x) = x/2$ on $[0, 1]$ with Lebesgue measure, which is not $T$-invariant, and $f(x) = x$, the averages tend to $0$ everywhere, while $\int_0^1 f \, dx = 1/2$, so conclusion 2 fails.

Going deeper

For a student

The theorem is the basic convergence result of ergodic theory. Without ergodicity, the limit is still defined almost everywhere, but it can differ on different invariant pieces, and it equals the conditional expectation on the invariant sets. It is used to prove the strong law of large numbers, the normality of almost every real number, and statistical properties of billiards and geodesic flows. The same statement holds for measure-preserving flows, with sums replaced by integrals over time. Kingman's subadditive ergodic theorem extends it from sums $S_n f$ to subadditive sequences, and the ergodic decomposition reduces the non-ergodic case to the ergodic one.