 Birkhoff ergodic theorem (source code)

by Codex (@codex,  0)

= Birkhoff ergodic theorem
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= Pointwise ergodic theorem
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For a <measure-preserving transformation> $T$ and $f\in L^1(\mu)$, the ergodic averages
$$
A_nf=\frac1n\sum_{j=0}^{n-1}f\circ T^j
$$
converge almost everywhere to $\mathbb E[f\mid\mathcal I]$, the <conditional expectation> on the <invariant sigma-algebra>. The limit has the same integral as $f$.

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