Conditional measures and conditional expectation; Rohlin's Disintegration Theorem

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.