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].