Maximal ergodic theorem

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

Suppose that ( X , B , μ ) is a probability space, that T : X → X is a (possibly noninvertible) measure-preserving transformation, and that f ∈ L 1 ( μ ) . Define f ∗ by

Then the maximal ergodic theorem states that

for any λ ∈ R.

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