Orthogonal Transition Kernels

Abstract

A transition kernel $(\mu_x)_{x \in X}$ between Polish spaces $X$ and $Y$ is called completely orthogonal if the $\mu_x$ are separated by the fibers of a Borel map $\varphi: Y \rightarrow X$. 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 $\sigma$-compact, $Y = \lim_\leftarrow Y_n$ and $(\mu_x)$ is orthogonality preserving and continuous in $x$ then there is even a strongly consistent sequence of statistics $\varphi_n: Y_n \rightarrow X$ for $(\mu_x)$.