Let 
and 
be measure spaces, let 
be the collection of all measurable rectangles contained in 
, and let 
be the premeasure defined
on 
by
 |
for 
.
The product measure 
is the Carathéodory extension of
![lambda:R->[0,infty]](/images/equations/ProductMeasure/Inline9.svg)
defined on the
sigma-algebra of 
-measurable subsets of 
where 
denotes the outer
measure induced by the premeasure 
on 
.
