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
 * $$ \int_{f^{*} > \lambda} f \, d\mu \ge \lambda \cdot \mu\{ f^{*} > \lambda\} $$

for any λ ∈ R.

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