Contents of Measure Theory, by D.H.Fremlin

Chapter 34: Liftings

341 The lifting theorem
Liftings and lower densities; strictly localizable spaces have lower densities; construction of a lifting from a density; complete strictly localizable spaces have liftings; liftings and Stone spaces.

342 Compact measure spaces
Inner regular measures; compact classes; compact and locally compact measures; perfect measures.

343 Realization of homomorphisms
Representing homomorphisms between measure algebras by functions; possible when target measure space is locally compact; countably separated measures and uniqueness of representing functions; the split interval; perfect measures.

344 Realization of automorphisms
Simultaneously representing groups of automorphisms of measure algebras by functions -- Stone spaces, countably separated measure spaces, measures on {0,1}I; characterization of Lebesgue measure as a measure space; strong homogeneity of usual measure on {0,1}I.

345 Translation-invariant liftings
Translation-invariant liftings on Rr and {0,1}I; there is no t.-i. Borel lifting on R.

346 Consistent liftings
Liftings of product measures which respect the product structure; translation-invariant liftings on {0,1}I; products of Maharam-type-homogeneous probability spaces; lower densities respecting product structures; consistent liftings; the Stone space of Lebesgue measure.

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Revised 23.12.10