The Radon-Nikodym theorem asserts that any finite complex measure 
that is absolutely continuous with respect
to a sigma-finite positive measure 
(such as Lebesgue measure
on 
)
is given by the integral of some 
-function 
,
 |
(1)
|
The function 
is like a density function for the measure.
A closely related theorem says that any complex measure 
decomposes into an absolutely
continuous measure 
and a singular measure 
. This is the Lebesgue
decomposition,
 |
(2)
|
One consequence of the Radon-Nikodym theorem is that any complex measure has a polar representation
 |
(3)
|
with 
.
