Contents of Measure Theory, by D.H.Fremlin

Chapter 55: Possible worlds

551 Forcing with quotient algebras
Measurable spaces with negligibles; associated forcing notions; representing names for members of {0,1}I; representing names for Baire sets in {0,1}I; the usual measure on {0,1}I; re-interpreting Baire sets in the forcing model; representing Baire measurable functions; representing measure algebras; iterated forcing; extending filters.

552 Random reals I
Random real forcing notions; calculating 2κ; b and d; preservation of outer measure; Sierpinski sets; cardinal functions of the usual measure on {0,1}λ; Carlson's theorem on extending measures; iterated random real forcing.

553 Random reals II
Rothberger's property; non-scattered compact sets; Haydon's property; rapid p-point ultrafilters; products of ccc partially ordered sets; Aronszajn trees; medial limits; universally measurable sets.

554 Cohen reals
Calculating 2κ; Lusin sets; precaliber pairs of measure algebras; Freese-Nation numbers; Borel liftings for Lebesgue measure.

555 Solovay's construction of real-valued-measurable cardinals
Measurable cardinals are quasi-measurable after ccc forcing, real-valued-measurable after random real forcing; Maharam-type-homogeneity; covering number of product measure; power set σ-quotient algebras can have countable centering number or Maharam type; supercompact cardinals and the normal measure axiom.

556 Forcing with Boolean subalgebras
Forcing names over a Boolean subalgebra; Boolean operations, ring homomorphisms; when the subalgebra is regularly embedded; upper bounds, suprema, saturation, Maharam type; quotient forcing; Dedekind completeness; L0; probability algebras; relatively independent subalgebras; strong law of large numbers; Dye's theorem; Kawada's theorem; the Dedekind completion of the asymptotic density algebra.

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Revised 13.11.10