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 $$f \in L^1(\mu)$$. Define $$f^*$$ by
 * $$f^* = \sup_{N=1}^\infty \frac1N \sum_{i=0}^{N-1} f \circ T^i. $$

Then the maximal ergodic theorem states that
 * $$ \int_{f^* > \lambda} f \,d\mu \ge \lambda \cdot \mu\{ f^* > \lambda\} $$

for any &lambda; &isin; R.

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