A Characterization of Orthogonal Transition Kernels

Abstract

A transition kernel $\mu = (\mu_y)_{y\in Y}$ between Polish spaces $X$ and $Y$ is completely orthogonal if there is a perfect statistic $\varphi: X \rightarrow Y$ for $\mu$, i.e. the fibers of the Borel map $\varphi$ separate the $\mu_y$. Equivalent properties are: a) orthogonal, finitely additive measures $p, q$ on $Y$ induce orthogonal mixtures $\mu^p, \mu^q$ on $X$; b) the Markov operator defined by $\mu$ is subjective on a certain class of Borel functions.