A polar representation of a complex measure 
is analogous to the polar representation
of a complex number as 
, where 
,
 |
(1)
|
The analog of absolute value is the total variation 
, and 
is replaced by a measurable
real-valued function 
. 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
 |
(3)
|
