The Jordan measure decomposition of a real measure 
expresses it as the difference of its positive and negative variations (Hunter 2011,
p. 73). 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
with respect to Lebesgue measure on 
, then
 |
(4)
|
where 
is a measurable real-valued
function with 
(Hunter 2011, pp. 77-78). The variations
and 
are also absolutely continuous with respect
to Lebesgue measure. They can be written as
 |
(5)
|
and
 |
(6)
|
where 
is the decomposition of 
into its positive part and
negative part.
The Jordan decomposition has a so-called minimum property. In particular, given any finite positive measure 
, the real measure 
has another decomposition
 |
(7)
|
The Jordan decomposition is minimal with respect to these changes. One way to say this is that any decomposition 
into finite positive
measures must have 
and 
.
