Contents of Measure Theory, by D.H.Fremlin

Chapter 25: Product measures

251 Finite products
Primitive and c.l.d. products; basic properties; Lebesgue measure on Rr+s as a product measure; products of direct sums and subspaces; c.l.d. versions.

252 Fubini's theorem
When ∫∫f (x,y)dxdy and ∫f (x,y)d(x,y) are equal; measures of ordinate sets; *the volume of a ball in Rr.

253 Tensor products
Bilinear operators; bilinear operators L1(μ)×L1(ν)→W and linear operators L1(μ×ν)→W; positive bilinear operators and the ordering of L1(μ×ν); conditional expectations; upper integrals.

254 Infinite products
Products of arbitrary families of probability spaces; basic properties; inverse-measure-preserving functions; usual measure on {0,1}I; {0,1}N isomorphic, as measure space, to [0,1]; subspaces of full outer measure; sets determined by coordinates in a subset of the index set; generalized associative law for products of measures; subproducts as image measures; factoring functions through subproducts; conditional expectations on subalgebras corresponding to subproducts.

255 Convolutions of functions
Shifts in R2 as measure space automorphisms; convolutions of functions on R; ∫h×(f*g)=∫h(x+y)f(x)g(y)d(x,y); f *(g*h)=(f *g)*h; ∥f*g1≤ ∥f ∥1g1; the groups Rr and ]-π,π].

256 Radon measures on Rr
Definition of Radon measures on Rr; completions of Borel measures; Lusin measurability; image measures; products of two Radon measures; semi-continuous functions.

257 Convolutions of measures
Convolution of totally finite Radon measures on Rr; ∫h d12)=∬h(x+y1(dx2(dy); ν1*(ν23)=(ν12)*ν3; convolutions and Radon-Nikodým derivatives.

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Revised 22.8.13