Conditional measures and conditional expectation; Rohlin's Disintegration Theorem
By David Simmons. 2012
80 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/4a4225ed5a7c0474d79d0a50ccfcf7cec15009c1
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Citations in the corpus
- Ledrappier-Young formula and exact dimensionality of self-affine measures (2015, unknown): “provided that the limit exists; see [42].”
- A proof of Furstenberg's conjecture on the intersections of ⨯p- and ⨯q-invariant sets (2016, background): “1090/memo/1195 [35] Simmons, David, Conditional measures and conditional expectation; {R}ohlin’s disintegration theorem, Discrete Contin.”