A polar representation of a complex measure 
expresses it as
 |
(1)
|
This is analogous to the polar representation of a complex number as 
,
where 
.
The analog of absolute value is the total
variation 
,
and 
is a measurable function with values in 
(Rudin 1987, pp. 124-125). Or sometimes
one writes 
with 
instead of 
.
More precisely, for any measurable set 
,
 |
(2)
|
where the integral is the Lebesgue integral. It is natural to extend the definition of the Lebesgue
integral to complex measures using the polar
representation. For a measurable function 
with values in 
and 
,
 |
(3)
|
