If 
is a real measure (i.e., a measure
that takes on real values), then one can decompose it according to where it is positive
and negative. The positive variation is defined by
 |
(1)
|
where 
is the total variation. Similarly, the negative
variation is
 |
(2)
|
Then the Jordan decomposition of 
is defined as
 |
(3)
|
When 
already is a positive measure then 
. More generally, if 
is absolutely continuous,
i.e.,
 |
(4)
|
then so are 
and 
.
The positive and negative variations can also be written as
 |
(5)
|
and
 |
(6)
|
where 
is the decomposition of 
into its positive and negative parts.
The Jordan decomposition has a so-called minimum property. In particular, given any positive measure 
, the measure 
has another decomposition
 |
(7)
|
The Jordan decomposition is minimal with respect to these changes. One way to say this is that any decomposition 
must have 
and 
.
