Orthogonal Transition Kernels

A transition kernel (μx)x ∈ Xbetween Polish spaces X and Y is called completely orthogonal if the μxare separated by the fibers of a Borel map φ: Y → X. It is orthogonality preserving if orthogonal measures on X induce orthogonal mixtures on Y. We give a von Neumann "type" isomorphism theorem for atomless completely orthogonal kernels, and a theorem and some counterexamples concerning the separation of two orthogonal measure convex sets of probability measures by a measurable set. These techniques yield three results on orthogonality preserving kernels: (1) They need not be completely orthogonal but (2) are uniformly orthogonal (in the sense of D. Maharam) and (3) if X is σ-compact,$Y = \lim_\leftarrow Y_n$and (μx) is orthogonality preserving and continuous in x then there is even a strongly consistent sequence of statistics φn: Yn→ X for (μx).