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Milds # 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 $f\in L^{1}(\mu ,\mathbb {R} )$ . 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

$\int _{f^{*}>\lambda }f\,d\mu \geq \lambda \cdot \mu \{f^{*}>\lambda \}$ for any λ ∈ R.

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