A measure 
is absolutely continuous with respect to another measure
if 
for every set with 
. This makes sense as long as 
is a positive measure,
such as Lebesgue measure, but 
can be any measure, possibly a complex
measure.
By the Radon-Nikodym theorem, this is equivalent to saying that
 |
where the integral is the Lebesgue integral, for some integrable function 
. The function 
is like a derivative, and is called the Radon-Nikodym
derivative 
.
The measure supported at 0 (
iff 
) is not absolutely continuous with respect to Lebesgue
measure, and is a singular measure.
