Given a complex measure 
, there exists a positive
measure denoted 
which measures the total variation of 
, also sometimes called simply "total variation."
In particular, 
on a subset 
is the largest sum of "variations" for any subdivision
of 
.
Roughly speaking, a total variation measure is an infinitesimal version of the absolute value.
More precisely,
 |
(1)
|
where the supremum is taken over all partitions 
of 
into measurable subsets 
.
Note that 
may not be the same as 
.
When 
already is a positive measure, then 
. More generally, if 
is absolutely continuous,
that is
 |
(2)
|
then so is 
,
and the total variation measure can be written as
 |
(3)
|
The total variation measure can be used to rewrite the original measure, in analogy to the norm of a complex number. The measure 
has a polar
representation
 |
(4)
|
with 
.
