Maximal ergodic theorem

The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics.
Suppose that formula_1 is a probability space, that formula_2 is a (possibly noninvertible) measure-preserving transformation, and that formula_3. Define formula_4 by
Then the maximal ergodic theorem states that
for any λ ∈ R.
This theorem is used to prove the point-wise ergodic theorem.