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)