Maximal ergodic theorem

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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} \frac{1}{N} \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.


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