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Maximal ergodic theorem

The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics.

Suppose that $\left(X, mathcal\left\{B\right\}, mu\right)$ is a probability space, that $T : X to X$ is a (possibly noninvertible) measure-preserving transformation, and that $f in L^1\left(mu\right)$. Define $f^*$ by

$f^* = sup_\left\{N=1ldotsinfty\right\} frac1N sum_\left\{i=0\right\}^\left\{N-1\right\} f circ T^i.$
Then the maximal ergodic theorem states that
$int_\left\{f^* > lambda\right\} f ,dmu ge lambda cdot mu\left\{ f^* > lambda\right\}$
for any λ ∈ R.

This theorem is used to prove the point-wise ergodic theorem.

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