Invariant measures and Cesàro summability

Howard Jacob Weiner

Vol. 25 (1968), No. 3, 621–629

DOI: 10.2140/pjm.1968.25.621

Abstract

It is known that if T is a one-to-one, measurable, invertible and nonsingular transformation on the unit interval with a σ-finite invariant measure, then its induced transformation T₁ on L₁ functions f is such that lim_n→∞1∕n∑ _k=1ⁿT₁^kf(x) exists. In this note, a counterexample is constructed which shows that the converse is false.

Mathematical Subject Classification

Primary: 28.70

Milestones

Received: 6 July 1967

Published: 1 June 1968