Some uniform ergodic inequalities in the nonmeasurable case (Q1268777)

The Hopf-Yosida-Kakutani maximal ergodic theorem [\textit{U. Krengel}, ``Ergodic theorems'' (1985; Zbl 0575.28009)] has as a consequence the so-called maximal ergodic inequality. In the paper under review, the author establishes some uniform and nonmeasurable versions of this classical inequality. Namely, one considers an arbitrary, not necessarily \(\sigma\)--finite measure space \((\Omega,{\mathcal A},\mu)\) and the associated space \({\mathcal L}_1\equiv {\mathcal L}_1(\Omega,{\mathcal A}, \mu)\) of all \(\mu\)--integrable functions \(f:\Omega\to {\mathbb R}\). A positive \({\mathcal L}_1\)--contraction is a linear operator \(T:{\mathcal L}_1\to {\mathcal L}_1\) such that \(Tf\geq 0\) \(\forall f\geq 0\), and \(\int_{\Omega}| Tf | d\mu \leq \int_{\Omega}| f | d\mu\), \(\forall f\in {\mathcal L}_1\). The author extends \(T\) naturally to \({\overline{\mathbb R}}^\Omega\), i.e., to the class of all (not necesarily measurable) functions on \(\Omega\). Then it is stated and proved a uniform and nonmeasurable version of the Hopf-Yosida-Kakutani maximal ergodic theorem. Moreover, this extension leads to technical improvements of the proofs and to remarkable generalizations of classical ergodic inequalities.