Some Uniform Ergodic Inequalities in the Nonmeasurable Case

Uniform and nonmeasurable versions of some classical ergodic inequalities (all of them going back to the Hopf–Yosida–Kakutani maximal ergodic theorem) are estab- lished. Usually, uniformity involves nonmeasurable suprema and all the technical difficulties arising from this. In the present paper, a simplification is achieved by extending the given operator (a positive L1-contraction) to the class of all (i.e., not necessarily measurable) functions on the underlying measure space. This not only leads to technical improvements and clarifications of the proofs, but also to remarkable generalizations of known results. In particular, it turns out that the “operator” under consideration need not even be an extension of an L1-contraction, but has only to fulfill some mild conditions such as positivity, super-additivity, and a certain contractivity property involving upper integrals.