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.

Title	ergodic theorem
Canonical name	ErgodicTheorem
Date of creation	2013-03-22 12:20:52
Last modified on	2013-03-22 12:20:52
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	11
Author	Koro (127)
Entry type	Theorem
Classification	msc 37A30
Classification	msc 47A35
Synonym	strong ergodic theorem
Synonym	Birkhoff ergodic theorem
Synonym	Birkhoff-Khinchin ergodic theorem
Related topic	ErgodicTransformation