The Radon-Nikodym theorem asserts that any absolutely continuous complex measure 
with respect to some positive measure 
(which could be Lebesgue
measure or Haar measure) 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 
.
