Representation of invariant measures

Abstract

sentation of invariant measures may be constructed, using the methods of Blum and Hanson, under broad enough conditions that we are able to show how to construct the representation in each of the cases mentioned above.Throughout this paper X will be a set, a z-algebra of subsets of X.(X, , ) will be called a probability space if is a nonnegative countably additive measure defined on such that (X)= 1.A transformation T :X---X will be called measurable if A e implies T-1A e .A measurable transformation T will be called measure-preserving (relative to (X, , )) if A e implies (A) (T-1A).Throughout, @ will be a set of measurable transformations of X into X.A probability measure on will be called invariant (relative to (X, , @) if A e , T e @ imply (A) (T-IA).Relative to (X, , @) we let be the set of invariant probability measures, and 1 the set of extreme points of .The convex set may not have any extreme points, but in the situations discussed in this paper, if is nonempty, then is also nonempty.In the following, 0 will be the a-algebra of measurable subsets invariant under the transformations in @, that is, A e 0 if and only irA e and for all Te@, A T-A. Ameasure ewill be

Locations

• Illinois Journal of Mathematics - View - PDF