Mean Ergodic Theorem

Theorem
Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system.

Let $\map {L^2_\C} \mu$ be the complex-valued $L^2$ space of $\mu$.

Let $U_T : \map {L^2_\C} \mu \to \map {L^2_\C} \mu$ of $T$ be the Koopman operator.

Let $I := \set {f \in \map {L^2_\C} \mu : \map {U_T} f = f}$.

Then for each $f \in \map {L^2_\C} \mu$:
 * $\ds \lim_{N \mathop \to \infty} \dfrac 1 N \sum_{n \mathop = 0}^{N - 1} \map {U_T^n} f = \map {P_T} f$

converges in the $L^2$-norm, where:
 * $U_T^n$ denotes the $n$ times composition of $U_T$
 * $P_T : \map {L^2_\C} \mu \to \map {L^2_\C} \mu$ denotes the orthogonal projection on the closed linear subspace $I$.

Proof
Recall $L^2$ space forms Hilbert space.

That is, $\map {L^2_\C} \mu$ is a Hilbert space with the $L^2$ inner product:
 * $\ds \innerprod f g := \int \overline f g \rd \mu$

Observe:
 * $I = \map \ker {U_T - \operatorname {id} }$

where $\operatorname {id}$ is the identity mapping on $\map {L^2_\C} \mu$.

Therefore $I$ is a closed linear subspace by Kernel of Bounded Linear Transformation is Closed Linear Subspace.

Lemma
By Direct Sum of Subspace and Orthocomplement, we have:
 * $\map {L^2_\C} \mu = \overline B \oplus I$

Thus it suffices to show that for all $f \in \overline B$:
 * $\ds \lim_{N \mathop \to \infty} \dfrac 1 N \sum_{n \mathop = 0}^{N - 1} \map {U_T^n} f = 0$

converges in the $L^2$-norm.

To this end, let $f \in \overline B$.

Let $\epsilon > 0$ be arbitrary.

Then there is a $g \in B$ such that:
 * $\norm {f - g} < \epsilon$

where $g = \map {U_T} h - h$ for a $g \in \map {L^2_\C} \mu$.

Thus for all $N \ge 2 \norm g \epsilon^{-1}$: