Contents of Measure Theory, by D.H.Fremlin

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 P1); notes on Lebesgue measure; the split interval.

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Revised 4.1.2018