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 
.
