Maximal Ergodic Theorem

Theorem

Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system.

Let $g : X \to \overline \R$ be a $\mu$-integrable function.

Let $\alpha \in \R$.

Let:

$\ds E_\alpha := \set {x \in X : \sup_{n \ge 1} \frac 1 n \sum_{i \mathop = 0}^{n-1} \map g {T^i x} > \alpha }$

Then:

$\ds \alpha \map \mu {E_\alpha} \le \int_{E_\alpha} g \rd \mu \le \int \size g \rd \mu$

Proof

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Sources

• 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory $2.6$ Pointwise Ergodic Theorem