Chapter 25: Product measures

251
Finite products

Primitive and c.l.d. products; basic properties; Lebesgue measure
on **R**^{r+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 **R**^{r}.

253
Tensor products

Bilinear operators; bilinear operators
*L*^{1}(μ)×*L*^{1}(ν)→*W* and linear operators
*L*^{1}(μ×ν)→*W*; positive bilinear operators and the
ordering of *L*^{1}(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 **R**^{2} 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 **R**^{r} and
]-π,π].

256
Radon measures on **R**^{r}

Definition of Radon measures on **R**^{r}; 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 **R**^{r};
∫*h d*(ν_{1}*ν_{2})=∬*h*(*x*+*y*)ν_{1}(*dx*)ν_{2}(*dy*);
ν_{1}*(ν_{2}*ν_{3})=(ν_{1}*ν_{2})*ν_{3}; convolutions and
Radon-Nikodým derivatives.

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Revised 22.8.13