ergodic theorem

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=1ldotsinfty} frac1N sum_{i=0}^{N-1} f circ T^i.
Then the maximal ergodic theorem states that
int_{f^* > lambda} f ,dmu ge lambda cdot mu{ f^* > lambda}
for any λ ∈ R.

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


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