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 $(\mu_x)$.
Publié le : 1983-11-14
Classification:
Orthogonal measures,
perfect statistics,
filters of countable type,
classification of kernels,
simultaneous Lebesgue decompositions,
60A10,
28A75
@article{1176993446,
author = {Mauldin, R. Daniel and Preiss, David and v. Weizsacker, Heinrich},
title = {Orthogonal Transition Kernels},
journal = {Ann. Probab.},
volume = {11},
number = {4},
year = {1983},
pages = { 970-988},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993446}
}
Mauldin, R. Daniel; Preiss, David; v. Weizsacker, Heinrich. Orthogonal Transition Kernels. Ann. Probab., Tome 11 (1983) no. 4, pp. 970-988. http://gdmltest.u-ga.fr/item/1176993446/