Product Measure
Consider two measure spaces
and 
.If a sequence of sets
that satisfies
exists, the product measure
is expressed as 
.Among all product measures
, the one given by
is called rectangular set.
All sets that can be represented as the sum of a finite number of rectangular sets form a finite additive class, and the smallest
-additive class that includes a finite additive class is called product Borel set and is expressed as
For a rectangular set
, the measure 
is defined as
and is denominated product measure.
The space of product measures is expressed as
A point
in 
is expressed as
Let
be a subset of 
.For a point
in 
, we have
.Similarly, we can write
.If
and 
are complete, the complete measure space is represented as
,and the product measure space is represented as
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