The Annals of Probability

Conditional Distributions and Orthogonal Measures

John P. Burgess and R. Daniel Mauldin

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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.

Article information

Ann. Probab., Volume 9, Number 5 (1981), 902-906.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60B05: Probability measures on topological spaces
Secondary: 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] 28A10: Real- or complex-valued set functions

Mutually singular measures conditional probability distribution


Burgess, John P.; Mauldin, R. Daniel. Conditional Distributions and Orthogonal Measures. Ann. Probab. 9 (1981), no. 5, 902--906. doi:10.1214/aop/1176994320.

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