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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Factorization of measures and normal conditional distributions

Authors: A. Maitra and S. Ramakrishnan
Journal: Proc. Amer. Math. Soc. 103 (1988), 1259-1267
MSC: Primary 60A10; Secondary 28D05
MathSciNet review: 955019
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Abstract: Let $ (Y,\mathcal{C},Q)$ be a probability space. If every probability measure $ R$ on $ {\mathcal{B}^1} \otimes \mathcal{C}$ with marginal $ Q$ on $ Y$ admits a factorization, where $ {\mathcal{B}^1}$ is the Borel $ \sigma $-field on the real line, $ Q$ must be perfect. Conversely if $ Q$ is perfect and $ \mathcal{C}$ is $ {\aleph _1}$-generated, then (a) for any measure $ R$ on $ \mathcal{A} \otimes \mathcal{C}$ with marginal $ Q$, where $ \mathcal{A}$ is any $ \sigma $-field of subsets of a set $ X$, there is a factorization; (b) for every tail-like sub-$ \sigma $-field $ \mathcal{D}$ of $ \mathcal{C}$, there is a normal conditional distribution given $ \mathcal{D}$. In special cases of interest, normal conditional distributions, satisfying additional desirable properties, are shown to exist.

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Keywords: Factorization, disintegration, conditional distribution, normal conditional distribution, perfect probability space, tail-like $ \sigma $-field, measure-preserving transformation, invariant $ \sigma $-field, symmetric $ \sigma $-field
Article copyright: © Copyright 1988 American Mathematical Society