A signed measure on a measurable space 
is a set function ![nu:A->[-infty,infty]](/images/equations/SignedMeasure/Inline2.svg)
that is countably
additive, vanishes on the empty set, and does not
take both 
and 
as values (Hunter 2011, pp. 70-71).
A signed measure is finite when all its values are finite. A finite signed measure is a real measure, equivalently a complex measure whose values are all real numbers (Rudin 1987, p. 119).
For a finite signed measure, the Jordan measure decomposition gives a unique representation
 |
where 
and 
are finite mutually singular positive
measures. Its total variation is 
, so 
(Hunter 2011, pp. 73-74). Conversely,
a signed measure with finite total variation is
finite.
