Some generalizations of the ergodic theorem

Abstract

Throughout this paper we shall suppose that denotes a set of elements x in which a Lebesgue measure is defined and that itself is measurable and has finite measure. A (1, 1) transformation T of into itself is called an equimeasure transformation if the transform T E of any measurable subset E of is measurable and has measure equal to that of E . Then, if f ( x ) is integrable in , it is plain that f ( Tx ) is also integrable and that