In General a Measure Preserving Transformation is Mixing

Abstract

this note I continue the study of topological properties of the group of measure preserving transformations begun in an earlier paper.' Using the methods and results of that paper I present the first proof of the old standing conjecture stated in the title.2 In general means of course that the exceptional set is of the first category in one of the usual topologies (the strong neighborhood topology) for measure preserving transformations. The principal new and quite surprising fact used in the proof is that for any almost nowhere periodic measure preserving transformation T (and a fortiori for any mixing T) the set of all conjugates of T(i.e. the set of all STSJ') is everywhere dense. It is this possibility of a dense conjugate class in a comparatively well behaved topological group (a rather natural generalization of the finite symmetric groups) that is contrary to naive intuition. Let G be the group of all measure preserving transformations of the unit interval.3 For any S e G, measurable set a, and positive number E, write