Maximal ergodic theorem explained

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

Suppose that

(X, l{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,R)} . Define f ^{*} byf ^{*} = \sup _{N\geq} (1)/(N) \sum ^{N -1} _{i =0} f \circ T ^{i.} Then the maximal ergodic theorem states that\int _{f ^{*} > λ} f d\mu \ge λ ⋅ \mu\{ f ^{*} > λ \} for any λ ∈ R.

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

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