Conditional Versions of the Strassen-Dudley Theorem

Let X be a random variable with values in a Polish space B and defined on a rich enough probability space (Ω, S, P). (This will be clarified later.) Let F and G be sub-σ-fields of S and let G(|G) be a regular conditional distribution on B, defined on (Ω, S, P) and measurable with respect to G.