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1. Tensor product of measures and Fubini theorem. Let (Aj, j, j) , j = 1, 2 , be two measure spaces. Recall that the new -algebra A1 A2
 

Summary: 1. Tensor product of measures and Fubini theorem. Let (Aj, j, j) ,
j = 1, 2 , be two measure spaces. Recall that the new -algebra A1 A2
with the unit element 1 2 is the -algebra generated by the direct
products of elements of A1,2 . That is,
A1 A2 = 12
({E1 E2 : E1,2 A1,2}) .
Our goal first will be to define a product measure on A1 A2 .
2. A measure defined on the -algebra A1 A2 , such that
(E1 E2) = 1(E1)2(E2) E1,2 A1,2
is called the (tensor) product of 1 and 2 . A somewhat surprising fact
is that in such general situation the product is not unique, although
always exists. We will denote any product measure by 1 2 . Thus
= 1 2
makes sence, but 1 2 = is a uniqueness statement (which might be
wrong).
3. We shall establish the existence of 1 2 by running the Caratheodory
maschine. We need to prepare the setting for it. In the following notice
the strong similarities with the construction of n
on Rn
.

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics