# ergodic theorem

Let $(X,\mathfrak{B},\mu)$ be a probability space, $f\in L^{1}(\mu)$, and $T\colon X\to X$ a measure preserving transformation. Birkhoff’s ergodic theorem (often called the pointwise or strong ergodic theorem) states that there exists $f^{*}\in L^{1}(\mu)$ such that

 $\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f(T^{k}x)=f^{*}(x)$

for almost all $x\in X$. Moreover, $f^{*}$ is $T$-invariant (i.e., $f^{*}\circ T=f^{*}$) almost everywhere and

 $\int f^{*}d\mu=\int fd\mu.$

In particular, if $T$ is ergodic then the $T$-invariance of $f^{*}$ implies that it is constant almost everywhere, and so this constant must be the integral of $f^{*}$; that is, if $T$ is ergodic, then

 $\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f(T^{k}x)=\int fd\mu$

for almost every $x$. This is often interpreted in the following way: for an ergodic transformation, the time average equals the space average almost surely.

Title ergodic theorem ErgodicTheorem 2013-03-22 12:20:52 2013-03-22 12:20:52 Koro (127) Koro (127) 11 Koro (127) Theorem msc 37A30 msc 47A35 strong ergodic theorem Birkhoff ergodic theorem Birkhoff-Khinchin ergodic theorem ErgodicTransformation