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October, 1981 Conditional Distributions and Orthogonal Measures
John P. Burgess, R. Daniel Mauldin
Ann. Probab. 9(5): 902-906 (October, 1981). DOI: 10.1214/aop/1176994320


It is shown that every family of mutually singular measures in a conditional probability distribution is countable or else there is a perfect set of measures which form a strongly orthogonal family. Theorem: Let $X$ and $Y$ be complete separable metric spaces and $\mu$ a conditional probability distribution on $X \times \mathscr{B}(Y)$. Then either (1) there is a nonempty compact perfect subset $P$ of $X$ and a Borel subset $D$ of $X \times Y$ so that if $x$ and $y$ are distinct elements of $P$, then $\mu(x, D_x) = 1, \mu(y, D_x) = 0$, and $D_x \cap D_y = \phi$ or else (2) if $K$ is a subset of $X$ so that $\{\mu(x, \cdot):x \in K\}$ is a pairwise orthogonal family, then $K$ is countable.


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John P. Burgess. R. Daniel Mauldin. "Conditional Distributions and Orthogonal Measures." Ann. Probab. 9 (5) 902 - 906, October, 1981.


Published: October, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0465.60005
MathSciNet: MR628885
Digital Object Identifier: 10.1214/aop/1176994320

Primary: 60B05
Secondary: 28A05 , 28A10

Keywords: conditional probability distribution , mutually singular measures

Rights: Copyright © 1981 Institute of Mathematical Statistics


Vol.9 • No. 5 • October, 1981
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