Chapter 45: Perfect measures, disintegrations and processes

451
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.

452
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.

453
Strong liftings

Strong and almost strong liftings; existence; on product spaces;
disintegrations of Radon measures over spaces with almost strong
liftings; Stone spaces; Losert's example.

454
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*.

455
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.

456
Gaussian distributions

Gaussian distributions and processes; covariance matrices,
correlation and independence; supports; universal Gaussian
distributions; cluster sets of *n*-dimensional processes;
τ-additivity.

457
Simultaneous extension of measures

Extending families of finitely additive functionals; Strassen's
theorem; extending families of measures; examples; the Wasserstein
metric.

458
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.

459
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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Revised 24.11.2017