Chapter 23: The Radon-Nikodým theorem

231
Countably additive functionals

Additive and countably additive functionals; Jordan and Hahn
decompositions.

232
The Radon-Nikodým theorem

Absolutely and truly continuous additive functionals; truly
continuous functionals are indefinite integrals; *the Lebesgue
decomposition of a countably additive functional.

233
Conditional expectations

σ-subalgebras; conditional expectations of integrable
functions; convex functions; Jensen's inequality.

234
Operations on measures

Inverse-measure-preserving functions; image measures; sums of
measures; indefinite-integral measures; ordering of measures.

235
Measurable transformations

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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Revised 19.1.16