Chapter 41: Topologies and measures I
411
Definitions
Topological, inner regular, τ-additive, outer regular, locally
finite, effectively locally finite, quasi-Radon, Radon, completion
regular, Baire, Borel and strictly positive measures; measurable and
almost continuous functions; self-supporting sets and supports of
measures; Stone spaces; Dieudonné's measure.
412
Inner regularity
Exhaustion; Baire measures; Borel measures on metrizable spaces;
completions and c.l.d. versions; complete locally determined
spaces; inverse-measure-preserving functions; subspaces; indefinite-integral measures;
products; outer regularity.
413
Inner measure constructions
Inner measures; constructing a measure from an inner measure; the
inner measure defined by a measure; complete locally determined
spaces; extension of functionals to measures; countably compact
classes; constructing measures dominating given functionals.
414
τ-additivity
Semi-continuous functions; supports; strict localizability;
subspace measures; regular topologies; density topologies; lifting
topologies.
415
Quasi-Radon measure spaces
Strict localizability; subspaces; regular topologies; hereditarily
Lindelöf spaces; products of separable metrizable spaces;
comparison and specification of quasi-Radon measures; construction
of quasi-Radon measures extending given functionals;
indefinite-integral measures; Lp spaces; Stone spaces.
416
Radon measure spaces
Radon and quasi-Radon measures; specification of Radon measures;
c.l.d. versions of Borel measures; locally compact topologies;
constructions of Radon measures extending or dominating given
functionals; additive functionals on Boolean algebras and Radon
measures on Stone spaces; subspaces; products; Stone spaces of
measure algebras; compact and perfect measures; representation of
homomorphisms of measure algebras.
417
τ-additive product measures
The product of two effectively locally finite τ-additive
measures; the product of many τ-additive probability measures;
Fubini's theorem; generalized associative law; measures on
subproducts as image measures; products of strictly positive
measures; quasi-Radon and Radon product measures; when `ordinary'
product measures are τ-additive; continuous functions and Baire
σ-algebras in product spaces.
418
Measurable functions and almost continuous functions
Measurable functions; into (separable) metrizable spaces; and image
measures; almost continuous functions; continuity, measurability,
image measures; expressing Radon measures as images of Radon
measures; Prokhorov's theorem on projective limits of Radon
measures; representing measurable functions into L0 spaces.
419
Examples
A nearly quasi-Radon measure; a Radon measure space in which the
Borel sets are inadequate; a nearly Radon measure; the Stone space
of the Lebesgue measure algebra; measures with domain
P(ω1); notes on Lebesgue measure; the split interval.
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