The maximal ergodic theorem is a

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...

ergodic theory Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...

, a discipline within mathematics. Suppose that

(X, \mathcal,\mu)

is a

probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...

, that

T : X\to X

is a (possibly noninvertible)

measure-preserving transformation In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special ca ...

, and that

f\in L^1(\mu,\mathbb)

. Define

f^*

by :

f^* = \sup_ \frac \sum_^ f \circ T^i.

Then the maximal ergodic theorem states that :

\int_ f \, d\mu \ge \lambda \cdot \mu\

for any λ ∈ R. This theorem is used to prove the point-wise

ergodic theorem Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...

References

* . Probability theorems Ergodic theory Theorems in dynamical systems {{chaos-stub