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# Maximal ergodic theorem

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 Title: Maximal ergodic theorem Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Maximal ergodic theorem

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

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

f^* = \sup_{N\geq 1} \frac1N \sum_{i=0}^{N-1} f \circ T^i.

Then the maximal ergodic theorem states that

\int_{f^* > \lambda} f \,d\mu \ge \lambda \cdot \mu\{ f^* > \lambda\}

for any λ ∈ R.

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