Disintegration Theorem

Michikazu Hirata 📧

November 2, 2023

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Abstract

We formalize mixture and disintegraion of measures. This entry is a formalization of Chapter 14.D of the book by Baccelli et.al. The main result is the disintegration theorem: let (X,ΣX) be a measurable space, (Y,ΣY) be a standard Borel space, ν be a σ-finite measure on X×Y, and νX be the marginal measure on X defined by νX(A)=ν(A×Y). Assume that νX is σ-finite, then there exists a probability kernel κ from X to Y such that ν(A×B)=∫Aκx(B)νX(dx),A∈ΣX,B∈ΣY. Such a probability kernel is unique νX-almost everywhere.

License

BSD License

Topics

Session Disintegration