Maximal Inequality for Positive Operators

Theorem

Let $\struct {X, \BB, \mu}$ be a probability space.

Let $\map {L^1} \mu$ be a real-valued $L^1$ space.

Let $U : \map {L^1} \mu \to \map {L^1} \mu$ be a positive linear operator, that is:

$\forall f \in \map {L^1} \mu : f \ge 0 \implies U f \ge 0$

Suppose:

$\norm U \le 1$

where $\norm \cdot$ denotes the operator norm.

Then, for all $N \in \N_{>0}$:

$\ds \int_{\set {F_N > 0} } f \rd \mu \ge 0$

where:

$F_N := \max \set {f_n : 0 \le n \le N}$

and:

$f_n := f + U f + \cdots + U^{n-1} f$

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