SOME UNIFORM ERGODIC INEQUALITIES IN THE NONMEASURABLE CASE

Uniform and nonmeasurable versions of some classical ergodic inequalit ies (all of them going back to the Hopf-Yosida-Kakutani maximal ergodi c theorem) are established. 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 ope rator (a positive L-1-contraction) to the class of all (i.e., not nece ssarily measurable) functions on the underlying measure space. This no t only leads to technical improvements and clarifications of the proof s, but also to remarkable generalizations of known results. In particu lar, it turns out that the ''operator'' under consideration need not e ven be an extension of an L-1-contraction, but has only to fulfill som e mild conditions such as positivity, super-additivity, and a certain contractivity property involving upper integrals. (C) 1998 Academic Pr ess.