ergodic theorem

Let $(X,𝔅,\mu )$ be a probability space, $f\in L^1(\mu )$, and $T:X\to X$ 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

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\sum _{k=0}^{n-1}f(T^{k}x)=f^{*}(x)$$

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

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\sum _{k=0}^{n-1}f(T^{k}x)=\int f𝑑\mu$$

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.