Kingman’s subadditive ergodic theorem
Let $(M,\mathcal{A},\mu )$ be a probability space^{}, and $f:M\to M$ be a measure preserving dynamical system^{}. let ${\varphi}_{n}:M\to \text{\mathbf{R}}$, $n\ge 1$ be a subadditive sequence of measurable functions^{}, such that ${\varphi}_{1}^{+}$ is integrable, where ${\varphi}_{1}^{+}=\mathrm{max}\{\varphi ,0\}$. Then, the sequence ${(\frac{{\varphi}_{n}}{n})}_{n}$ converges $\mu $ almost everywhere to a function $\varphi :M\to [\mathrm{\infty},\mathrm{\infty})$ such that:

${\varphi}^{+}$ is integrable

$\varphi $ is $f$ invariant, that is, $\varphi (f(x))=\varphi (x)$ for $\mu $ almost all $x$, and

$$\int \varphi \mathit{d}\mu =\underset{n}{lim}\frac{1}{n}\int {\varphi}_{n}\mathit{d}\mu =\underset{n}{inf}\frac{1}{n}\int {\varphi}_{n}\mathit{d}\mu \in [\mathrm{\infty},\mathrm{\infty})$$
The fact that the limit equals the infimum is a consequence of the fact that the sequence $\int {\varphi}_{n}\mathit{d}\mu $ is a subadditive sequence and Fekete’s subadditive lemma.
A superadditive version of the theorem also exists. Given a superadditive sequence ${\phi}_{n}$, 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 theorem^{} from Kingman’s subadditive ergodic theorem.
Title  Kingman’s subadditive ergodic theorem 

Canonical name  KingmansSubadditiveErgodicTheorem 
Date of creation  20140318 14:34:03 
Last modified on  20140318 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 