Kingman’s subadditive ergodic theorem

Let (M,𝒜,μ) be a probability spaceMathworldPlanetmath, and f:MM be a measure preserving dynamical systemMathworldPlanetmathPlanetmath. let ϕn:M𝐑, n1 be a subadditive sequence of measurable functionsMathworldPlanetmath, such that ϕ1+ is integrable, where ϕ1+=max{ϕ,0}. Then, the sequence (ϕnn)n converges μ almost everywhere to a function ϕ:M[-,) such that:

  • ϕ+ is integrable

  • ϕ is f invariant, that is, ϕ(f(x))=ϕ(x) for μ almost all x, and

  • ϕ𝑑μ=limn1nϕn𝑑μ=infn1nϕn𝑑μ[-,)

The fact that the limit equals the infimum is a consequence of the fact that the sequence ϕn𝑑μ is a subadditive sequence and Fekete’s subadditive lemma.

A superadditive version of the theorem also exists. Given a superadditive sequence φn, then the symmetric sequence is subadditive and we may apply the original version of the theorem.

Every additive sequence is subadditive. As a consequence, one can prove the Birkhoff ergodic theoremMathworldPlanetmath from Kingman’s subadditive ergodic theorem.

Title Kingman’s subadditive ergodic theorem
Canonical name KingmansSubadditiveErgodicTheorem
Date of creation 2014-03-18 14:34:03
Last modified on 2014-03-18 14:34:03
Owner Filipe (28191)
Last modified by Filipe (28191)
Numerical id 5
Author Filipe (28191)
Entry type Theorem
Related topic birkhoff ergodic theorem