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Summary: 1. Tensor product of measures and Fubini theorem. Let (Aj, j, µj) ,
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 = 1×2
({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) = µ1(E1)µ2(E2) E1,2 A1,2
is called the (tensor) product of µ1 and µ2 . 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 µ1 µ2 . Thus
= µ1 µ2
makes sence, but µ1 µ2 = is a uniqueness statement (which might be
wrong).
3. We shall establish the existence of µ1 µ2 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
.
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