Realizing a weak solution on a probability space

Let be a Borel application, v a given probability on , and [mu] a weak solution of the stochastic equation [mu]T-1=v. With ([Omega],,P) a probability space, and w:[Omega]-->Y a r.v. such that Pw-1=v, it is of interest to know when there is a r.v.x:[Omega]-->X such that Px-1=[mu] and Tx=w a.s. P. Such a r.v. is said to realize [mu] on ([Omega],,P) for w, and to factor w through T. It is known that if [mu] 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 [mu]-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:[Omega]-->R,a measure isomorphism h, and a decomposition (modP)[Omega]= E0[union or logical sum]E1[union or logical sum]E2[union or logical sum]..., with E0 conditionally w-1-atomless in , and En>0 disjoint conditional w-1-atoms in , such that 1. (i) . 2. (ii) 3. (iii) Under (PE0)/P(E0),f z.sfnc;E0is uniformly distributed on [0,1], and independent of E0[union or logical sum] w-1 4. (iv) . The atomless part E0 may be negligible, or there may be no atoms, but not both; [mu] 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.