A Note on Conservative Transformations and the Recurrence Theorem

Abstract

Let X be a space and &3 a a-field of measurable subsets of X, that is a class of closed under countable set-operations, with X C S. Let j be a class of of 03; we may call elements of .6 sets of measure Assume that a multi-valued transformation T is defined on X, i. e., a functioin which lets correspond to each point x C X a non-enipty set Tx C X. The empty set is denoted ?. T-1A is the set of points x such that Tx n A 0. The transformation T will be always assumed measurable, that is such that T-193 C S5. We shall say that T is conservative if A C A whenever a set A C 3 is disjoint with all T-tA, i = 1, 2, * -. It has been proved by ilalmos ([1], pp. 736-738, see also L2], p. 13), that any power Tn of a conservative transformation T is again conservative. Halmos assumes that T is singlevalued, so that Tx is a single point for each x, that T is one-to-one and that T-1 is measurable. The proof, based on an argument of E. Hopf ([3],-p. 391 ). is rather difficult. We propose here a simple proof of this result, which does not make use of the above assumptions. A version of the recurrence theorem extended in the same sense is then derived (compare [1], p. 747), with a corollary about a subsequence of the sequence T-nA.