TY - JOUR
AU1 - Matubara, Nozomu
AB - NOZOMU MATUBARA (Received Aug. 1, 1968) 1. Introduction Let (X, p) be a compact metric space, G a compact topological trans- formation group of X whose operation is effective and ~ a Borel z-field of X, that is, a countably additive class generated by all compact sets of X (or, equivalently, all open sets of X). A probability measure /2 on (X, ~) is called "invariant probability measure on (X, ~5)" if and only if /2(B)--/2(gB), g 6 G, B E ~5. Denote the family of all invariant probability measures on (X, ~) by ~. An invariant probability measure ~ ~ ~ is called "ergodic" if and only if ,(A) (1--u(A))=0, A E ~I where ~I={A; A E ~3, gA=A for all g E G} (an invariant sub-z-field). The family of all ergodic (invariant*) probability measures will be denoted by ~(c~). Now ~gt is a convex set, that is, for any /2,/E ~t ./2(.)+(1-.)/(.) e (o<.<1). plays an important r61e for ~Zt. Under certain conditions, 92 is found to consist of all extremal points of ~g}. (~ ~ ~ is said to be an extremal point of 93t when and only when ~ cannot be described as with [2,/2'6 
TI - On ergodic probability measures
JF - Annals of the Institute of Statistical Mathematics
DO - 10.1007/BF02532241
DA - 2006-11-17
UR - https://www.deepdyve.com/lp/springer-journals/on-ergodic-probability-measures-ONTzWvzqDi
SP - 175
EP - 183
VL - 21
IS - 1
DP - DeepDyve
ER -