Some Uniform Ergodic Inequalities in the Nonmeasurable Case

✍ Scribed by Klaus Ziegler

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 214 KB
• Volume: 154
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3216

✦ Synopsis

Uniform and nonmeasurable versions of some classical ergodic inequalities (all of them going back to the Hopf Yosida Kakutani maximal ergodic 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 operator (a positive L 1 -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 L 1 -contraction, but has only to fulfill some mild conditions such as positivity, super-additivity, and a certain contractivity property involving upper integrals.