Kingman’s subadditive ergodic theorem

Let $(M,𝒜,\mu )$ be a probability space, and $f:M\to M$ be a measure preserving dynamical system. let ${\varphi }_n:M\to \text{𝐑}$, $n\ge 1$ be a subadditive sequence of measurable functions, such that ${\varphi }_1^{+}$ is integrable, where ${\varphi }_1^{+}=\max \{\varphi ,0\}$. 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

• $$\int \varphi 𝑑\mu =\underset{n}{lim}\frac{1}{n}\int {\varphi }_n𝑑\mu =\underset{n}{inf}\frac{1}{n}\int {\varphi }_n𝑑\mu \in [-\mathrm{\infty },\mathrm{\infty })$$

The fact that the limit equals the infimum is a consequence of the fact that the sequence $\int {\varphi }_n𝑑\mu$ 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.