Realizing a weak solution on a probability space

AbstractLet T:(X,X)→(Y,Y) be a Borel application, v a given probability on Y, and μ a weak solution of the stochastic equation μT-1=v. With (Ω,F,P) a probability space, and w:Ω→Y a r.v. such that Pw-1=v, it is of interest to know when there is a r.v.x:Ω→X such that Px-1=μ and Tx=w a.s. |P|. Such a r.v. is said to realize μ on (Ω,F,P) for w, and to factor w through T. It is known that if μ is strong, or if the probability space is rich enough, then such a “realization” x exists; however, examples indicate that when T is injective on no set of full μ-measure, then there need be no such x. We give a n.a.s. condition for the existence of such a “realization” x' There must exist a r.v. f:Ω→R,a measure isomorphism h, and a decomposition (modP)Ω= E0∪E1∪E2∪…, with E0 conditionally w-1Y-atomless in w-1Yv f-1R, and En>0 disjoint conditional w-1Y-atoms in w-1Yv f-1R, such that 1.(i) h:(Ω, w-1Yv f-1R)→(μ,F).2.(ii) hw-1B=T-1B,BϵY3.(iii) Under (P|E0)P(E0),f z.sfnc;E0is uniformly distributed on [0,1], and independent of E0∪ w-14.(iv) f=n+12on En>0.The atomless part E0 may be negligible, or there may be no atoms, but not both; μ is strong ift there is one atom of full measure. Intuitively, the r.v. f represents the extra information in x over w, lost in the application T