ergodic theorem


Let (X,𝔅,μ) be a probability spaceMathworldPlanetmath, f∈L1⁢(μ), and T:X→X a measure preserving transformation. Birkhoff’s ergodic theorem (often called the pointwise or strong ergodic theorem) states that there exists f*∈L1⁢(μ) such that

limn→∞⁡1n⁢∑k=0n-1f⁢(Tk⁢x)=f*⁢(x)

for almost all x∈X. Moreover, f* is T-invariant (i.e., f*∘T=f*) almost everywhere and

∫f*⁢𝑑μ=∫f⁢𝑑μ.

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

limn→∞⁡1n⁢∑k=0n-1f⁢(Tk⁢x)=∫f⁢𝑑μ

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