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Kingman’s subadditive ergodic theorem
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.
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