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*g∥1≤ ∥f ∥1∥g∥1; 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 d(ν1*ν2)=∬h(x+y)ν1(dx)ν2(dy);
ν1*(ν2*ν3)=(ν1*ν2)*ν3; convolutions and
Radon-Nikodým derivatives.
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