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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, 2012, 32 (7) : 2565-2582. doi: 10.3934/dcds.2012.32.2565
References:
[1]

H. Bergström, "Weak Convergence of Measures,'' Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982.  Google Scholar

[2]

D. Cohn, "Measure Theory,'' Reprint of the 1980 original, Birkhäuser Boston, Inc., Boston, MA, 1993.  Google Scholar

[3]

G. de Rham, "Differentiable Manifolds. Forms, Currents, Harmonic Forms,'' Translated from the French by F. R. Smith, With an introduction by S. S. Chern, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 266, Springer-Verlag, Berlin, 1984.  Google Scholar

[4]

H. Federer, "Geometric Measure Theory,'' Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[5]

C. Hsiung, "A First Course in Differential Geometry,'' Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1981.  Google Scholar

[6]

D. Maharam, On the planar representation of a measurable subfield, in "Measure Theory, Oberwolfach 1983" (Oberwolfach, 1983), Lecture Notes in Math., 1089, Springer, Berlin, (1984), 47-57.  Google Scholar

[7]

V. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 1952 (1952), 55 pp.  Google Scholar

[8]

S. Srivastava, "A Course on Borel Sets,'' Graduate Texts in Mathematics, 180, Springer-Verlag, New York, 1998.  Google Scholar

[9]

S. Willard, "General Topology,'' Reprint of the 1970 original [Addison-Wesley, Reading, MA; MR0264581], Dover Publications, Inc., Mineola, NY, 2004.  Google Scholar

show all references

References:
[1]

H. Bergström, "Weak Convergence of Measures,'' Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982.  Google Scholar

[2]

D. Cohn, "Measure Theory,'' Reprint of the 1980 original, Birkhäuser Boston, Inc., Boston, MA, 1993.  Google Scholar

[3]

G. de Rham, "Differentiable Manifolds. Forms, Currents, Harmonic Forms,'' Translated from the French by F. R. Smith, With an introduction by S. S. Chern, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 266, Springer-Verlag, Berlin, 1984.  Google Scholar

[4]

H. Federer, "Geometric Measure Theory,'' Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[5]

C. Hsiung, "A First Course in Differential Geometry,'' Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1981.  Google Scholar

[6]

D. Maharam, On the planar representation of a measurable subfield, in "Measure Theory, Oberwolfach 1983" (Oberwolfach, 1983), Lecture Notes in Math., 1089, Springer, Berlin, (1984), 47-57.  Google Scholar

[7]

V. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 1952 (1952), 55 pp.  Google Scholar

[8]

S. Srivastava, "A Course on Borel Sets,'' Graduate Texts in Mathematics, 180, Springer-Verlag, New York, 1998.  Google Scholar

[9]

S. Willard, "General Topology,'' Reprint of the 1970 original [Addison-Wesley, Reading, MA; MR0264581], Dover Publications, Inc., Mineola, NY, 2004.  Google Scholar

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