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# 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.

## Mathematics Subject Classification

37A30*no label found*47A35

*no label found*

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