An absolutely continuous measure 
with respect to a measure 
is one for which 
for every measurable
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.
If 
is a sigma-finite positive measure and 
is a finite complex measure,
the Radon-Nikodym theorem makes this 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 
. Here, sigma-finiteness means
that the underlying space is a countable union of measurable
sets of finite 
-measure. The finiteness assumption on 
includes finite positive
measures and finite signed measures.
The measure supported at 0 (
iff 
) is not absolutely continuous with respect to Lebesgue
measure, and is a singular measure.
