Von Neumann’s ergodic theorem

Let $U:H\to H$ be an isometry in a Hilbert space $H$. Consider the subspace $I(U)=\{v\in H:Uv=v\}$, called the space of invariant vectors. Denote by $P$ the orthogonal projection over the subspace $I(U)$. Then,

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\sum _{j=0}^{n-1}{U}^j(v)=P(v),\forall v\in H$$

This general theorem for Hilbert spaces can be used to obtain an ergodic theorem for the $L^2(\mu )$ space by taking $H$ to be the $L^2(\mu )$ space, and $U$ to be the composition operator (also called Koopman operator) associated to a transformation $f:M\to M$ that preserves a measure $\mu$, i.e., $U_f(\psi )=\psi \circ f$, where $\psi :M\to \text{𝐑}$. The space of invariant functions is the set of functions $\psi$ such that $\psi \circ f=\psi$ almost everywhere. For any $\psi \in L^2(\mu )$, the sequence:

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\sum _{j=0}^{n-1}\psi \circ f^j$$

converges in $L^2(\mu )$ to the orthogonal projection $\tilde{\psi }$ of the function $\psi$ over the space of invariant functions.

The $L^2(\mu )$ version of the ergodic theorem for Hilbert spaces can be derived directly from the more general Birkhoff ergodic theorem, which asserts pointwise convergence instead of convergence in $L^2(\mu )$. Actually, from Birkhoff ergodic theorem one can derive a version of the ergodic theorem where convergence in $L^p(\mu )$ holds, for any $p>1$.