# American Institute of Mathematical Sciences

July  2012, 32(7): 2565-2582. doi: 10.3934/dcds.2012.32.2565

## Conditional measures and conditional expectation; Rohlin's Disintegration Theorem

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

Received  May 2011 Revised  June 2011 Published  March 2012

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.
Citation: David Simmons. Conditional measures and conditional expectation; Rohlin's Disintegration Theorem. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2565-2582. doi: 10.3934/dcds.2012.32.2565
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