Conditional measures and conditional expectation; Rohlin's Disintegration Theorem

David Simmons

doi:10.3934/dcds.2012.32.2565

Article Contents

2012, Volume 32, Issue 7: 2565-2582. Doi: 10.3934/dcds.2012.32.2565

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Conditional measures and conditional expectation; Rohlin's Disintegration Theorem

David Simmons^1,

1.
Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, TX 76203-1430

Received: April 30, 2011

Revised: May 31, 2011

Published: June 2012

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Abstract

The purpose of this paper is to give a clean formulation and proof of Rohlin's Disintegration Theorem [7]. Another (possible) proof can be found in [6]. Note also that our statement of Rohlin's Disintegration Theorem (Theorem 2.1) is more general than the statement in either [7] or [6] in that $X$ is allowed to be any universally measurable space, and $Y$ is allowed to be any subspace of standard Borel space.
Sections 1 - 4 contain the statement and proof of Rohlin's Theorem. Sections 5 - 7 give a generalization of Rohlin's Theorem to the category of $\sigma$-finite measure spaces with absolutely continuous morphisms. Section 8 gives a less general but more powerful version of Rohlin's Theorem in the category of smooth measures on $C^1$ manifolds. Section 9 is an appendix which contains proofs of facts used throughout the paper.

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Mathematics Subject Classification: Primary: 28A50, 28C15; Secondary: 28C05.

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References

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