Chapter 45: Perfect measures, disintegrations and processes
Perfect, compact and countably compact measures
Basic properties of the three classes; subspaces, completions, c.l.d. versions, products; measurable functions from compact measure spaces to metrizable spaces; *weakly α-favourable spaces.
Integration and disintegration of measures
Integrating families of measures; τ-additive and Radon measures; disintegrations and regular conditional probabilities; disintegrating countably compact measures; disintegrating Radon measures; *images of countably compact measures.
Strong and almost strong liftings; existence; on product spaces; disintegrations of Radon measures over spaces with almost strong liftings; Stone spaces; Losert's example.
Measures on product spaces
Perfect, compact and countably compact measures on product spaces; extension of finitely additive functions with perfect countably additive marginals; Kolmogorov's extension theorem; measures defined from conditional distributions; distributions of random processes; measures on C(T) for Polish T.
Markov and Lévy processes
Realization of a Markov process with given conditional distributions; the Markov property for stopping times taking countably many values -- disintegrations and conditional expectations; Radon conditional distributions; narrowly continuous and uniformly time-continuous systems of conditional distributions; càdlàg and càllàl functions; extending the distribution of a process to a Radon measure; when the subspace measure on the càdlàg functions is quasi-Radon; general stopping times, hitting times; the strong Markov property; independent increments, Lévy processes; expressing the strong Markov property with an inverse-measure-preserving function.
Gaussian distributions and processes; covariance matrices, correlation and independence; supports; universal Gaussian distributions; cluster sets of n-dimensional processes; τ-additivity.
Simultaneous extension of measures
Extending families of finitely additive functionals; Strassen's theorem; extending families of measures; examples; the Wasserstein metric.
Relative independence and relative products
Relatively independent algebras of measurable sets; relative distributions and relatively independent random variables; relatively independent subalgebras of a probability algebra; relative free products of probability algebras; relative products of probability spaces; existence of relative products.
Symmetric measures and exchangeable random variables
Exchangeable families of inverse-measure-preserving functions; De Finetti's theorem; countably compact symmetric measures on product spaces disintegrate into product measures; symmetric quasi-Radon measures; other actions of symmetric groups.
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