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# Von Neumann’s ergodic theorem

Let $U:H\rightarrow H$ 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,

$\lim_{{n\rightarrow\infty}}\frac{1}{n}\sum_{{j=0}}^{{n-1}}U^{j}(v)=P(v),% \forall v\in H$ |

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 $f:M\rightarrow M$ that preserves a measure $\mu$, i.e., $U_{f}(\psi)=\psi\circ f$, where $\psi:M\rightarrow\textbf{R}$. 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:

$\lim_{{n\rightarrow\infty}}\frac{1}{n}\sum_{{j=0}}^{{n-1}}\psi\circ f^{j}$ |

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 $p>1$.

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