Absolutely continuous measures
Suppose that on the measurable space there are given two measures
and
(cf. also Measure). One says that
is absolutely continuous with respect to
(denoted
) if
implies
for any set
. One also says that
dominates
. If the measure
is finite (i.e.
), then
if and only if for any
there exists a
such that
whenever
.
The Radon–Nikodým theorem says that if and
are
-finite measures and
, then there exists a
-integrable non-negative function
(a density, cf. also Integrable function), called the Radon–Nikodým derivative, such that
. Two such densities
and
may differ only on a null set (see Measure), i.e.
. An example of a density (with respect to the Lebesgue measure on the interval, i.e. the length) is the function
, where
is the sequence of all rational numbers in this interval.
The measure is -finite if
is the union of a countable family of sets with finite measure.
Given a reference measure on
, any measure may be decomposed into a sum of
and
with
and
, i.e. an absolutely continuous and a singular part. This is called the Lebesgue decomposition.
A set of non-zero measure that has no subsets of smaller, but still positive, measure is called an atom of the measure. It is a common mistake to claim that the singular part of a measure must be concentrated on points which are atoms. A singular measure may be atomless, as is shown by the measure concentrated on the standard Cantor set which puts zero on each gap of the set and on the intersection of the set with the interval of generation
.
When some canonical measure on
is fixed (as the Lebesgue measure on
or its subsets or, more generally, the Haar measure on a topological group), one says that
is absolutely continuous on
, meaning that
.
Two measures which are mutually absolutely continuous are called equivalent.
See also Absolute continuity.
References
[a1] | H.L. Royden, "Real analysis" , Macmillan (1968) |
[a2] | E. Hewitt, K. Stromberg, "Real and abstract analysis" , Springer (1965) |
Absolutely continuous measures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolutely_continuous_measures&oldid=14254