Chapter 23: The Radon-Nikodým theorem
Countably additive functionals
Additive and countably additive functionals; Jordan and Hahn decompositions.
The Radon-Nikodým theorem
Absolutely and truly continuous additive functionals; truly continuous functionals are indefinite integrals; *the Lebesgue decomposition of a countably additive functional.
σ-subalgebras; conditional expectations of integrable functions; convex functions; Jensen's inequality.
Operations on measures
Inverse-measure-preserving functions; image measures; sums of measures; indefinite-integral measures; ordering of measures.
The formula ∫g(y)ν(dy)=∫J(x)g(φ(x))μ(dx); detailed conditions of applicability; inverse-measure-preserving functions; the image measure catastrophe; using the Radon-Nikodým theorem.
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