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

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, 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.
Keywords: differential forms., conditional measures, linear functionals, Disintegration.
Mathematics Subject Classification: Primary: 28A50, 28C15; Secondary: 28C0.
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
References:
[1]

H. Bergström, "Weak Convergence of Measures,'', Probability and Mathematical Statistics, (1982).   Google Scholar

[2]

D. Cohn, "Measure Theory,'', Reprint of the 1980 original, (1980).   Google Scholar

[3]

G. de Rham, "Differentiable Manifolds. Forms, Currents, Harmonic Forms,'', Translated from the French by F. R. Smith, 266 (1984).   Google Scholar

[4]

H. Federer, "Geometric Measure Theory,'', Die Grundlehren der mathematischen Wissenschaften, (1969).   Google Scholar

[5]

C. Hsiung, "A First Course in Differential Geometry,'', Pure and Applied Mathematics, (1981).   Google Scholar

[6]

D. Maharam, On the planar representation of a measurable subfield,, in, 1089 (1984), 47.   Google Scholar

[7]

V. Rohlin, On the fundamental ideas of measure theory,, Amer. Math. Soc. Translation, 1952 (1952).   Google Scholar

[8]

S. Srivastava, "A Course on Borel Sets,'', Graduate Texts in Mathematics, 180 (1998).   Google Scholar

[9]

S. Willard, "General Topology,'', Reprint of the 1970 original [Addison-Wesley, (1970).   Google Scholar

show all references

References:
[1]

H. Bergström, "Weak Convergence of Measures,'', Probability and Mathematical Statistics, (1982).   Google Scholar

[2]

D. Cohn, "Measure Theory,'', Reprint of the 1980 original, (1980).   Google Scholar

[3]

G. de Rham, "Differentiable Manifolds. Forms, Currents, Harmonic Forms,'', Translated from the French by F. R. Smith, 266 (1984).   Google Scholar

[4]

H. Federer, "Geometric Measure Theory,'', Die Grundlehren der mathematischen Wissenschaften, (1969).   Google Scholar

[5]

C. Hsiung, "A First Course in Differential Geometry,'', Pure and Applied Mathematics, (1981).   Google Scholar

[6]

D. Maharam, On the planar representation of a measurable subfield,, in, 1089 (1984), 47.   Google Scholar

[7]

V. Rohlin, On the fundamental ideas of measure theory,, Amer. Math. Soc. Translation, 1952 (1952).   Google Scholar

[8]

S. Srivastava, "A Course on Borel Sets,'', Graduate Texts in Mathematics, 180 (1998).   Google Scholar

[9]

S. Willard, "General Topology,'', Reprint of the 1970 original [Addison-Wesley, (1970).   Google Scholar

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