Groups of automorphisms of Borel spaces 论文

摘要

1. Introduction.The main object of the present paper is to describe a generalization of some aspects of the classical von Neumann-Krylov-Bogoljubov theory of the decomposition of a suitably restricted Borel space into its so-called ergodic parts relative to a flow [1 ; 13; 18; 19].Our main results deal with locally compact groups of transformations acting measurably on sufficiently smooth Borel spaces, and lead to a fairly detailed picture of the action of such groups of automorphisms.The organization of the present paper is as follows.In §2 we summarize the basic material concerning Borel spaces which is pertinent to our subsequent discussions.In §3 we introduce the notion of a Borel G-space and examine the intimate relation between Borel G-spaces and topological G-spaces.We obtain, among other things, generalizations of well-known results of Mackey [17] and Ambrose-Kakutani [2].In §4 we take up the study of a standard Borel G-space.We show that any standard Borel G-space can be decomposed, in an essentially unique fashion, into disjoint invariant Borel sets on each of which the group acts uniquely ergodically.This decomposition leads to canonical forms for standard Borel G-spaces and to representations of invariant measures as integrals of ergodic measures.In §5 we study the convex set of all invariant measures on an arbitrary Borel G-space.We prove that every invariant measure is uniquely determined by its values on the cr-algebra of invariant sets and that the set of invariant measures is isomorphic to the set of all measures on some, possibly abstract, rr-algebra.In §6 we give a number of examples and make a few remarks supplementing the preceding discussions.