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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Factorization of measures and normal conditional distributions
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by A. Maitra and S. Ramakrishnan PDF
Proc. Amer. Math. Soc. 103 (1988), 1259-1267 Request permission

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.
References
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 103 (1988), 1259-1267
  • MSC: Primary 60A10; Secondary 28D05
  • DOI: https://doi.org/10.1090/S0002-9939-1988-0955019-3
  • MathSciNet review: 955019