The product measure 
of sigma-finite measure spaces 
and 
is the unique measure
on the product sigma-algebra 
satisfying
 |
for every measurable rectangle 
with 
and 
, using the convention 
(Hunter 2011, pp. 58-60). The measurable
rectangles form a semiring: they include the empty set,
are closed under intersection, and the set
difference of two is a finite union
of pairwise disjoint measurable
rectangles. They generate 
. The rectangle formula
extends by finite additivity to a premeasure
on the algebra of sets consisting of finite unions of measurable rectangles.
Its Carathéodory extension, restricted
to 
, is the product measure.
