Maximal ergodic theorem

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

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

${\displaystyle f^{*}=\sup _{N\geq 1}{\frac {1}{N}}\sum _{i=0}^{N-1}f\circ T^{i}.}$

Then the maximal ergodic theorem states that

${\displaystyle \int _{f^{*}>\lambda }f\,d\mu \geq \lambda \cdot \mu \{f^{*}>\lambda \}}$

for any λ ∈ R.

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