Let $(M,\mathcal{A},\mu)$ be a probability space, and $f:M\rightarrow M$ be a measure preserving dynamical system. let $\phi_{n}:M\rightarrow\textbf{R}$, $n\geq 1$ be a subadditive sequence of measurable functions, such that $\phi_{1}^{+}$ is integrable, where $\phi_{1}^{+}=\max\{\phi,0\}$. Then, the sequence $(\frac{\phi_{n}}{n})_{n}$ converges $\mu$ almost everywhere to a function $\phi:M\rightarrow[-\infty,\infty)$ such that:

• $\phi^{+}$ is integrable

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

•  $\int\phi d\mu=\lim_{n}\frac{1}{n}\int\phi_{n}d\mu=\inf_{n}\frac{1}{n}\int\phi_% {n}d\mu\in[-\infty,\infty)$

The fact that the limit equals the infimum is a consequence of the fact that the sequence $\int\phi_{n}d\mu$ is a subadditive sequence and Fekete’s subadditive lemma.

A superadditive version of the theorem also exists. Given a superadditive sequence $\varphi_{n}$, 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 KingmansSubadditiveErgodicTheorem 2014-03-18 14:34:03 2014-03-18 14:34:03 Filipe (28191) Filipe (28191) 5 Filipe (28191) Theorem birkhoff ergodic theorem