In mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if its support is at most a countable set. Note that the support need not be a discrete set. Geometrically, a discrete measure (on the real line, with respect to Lebesgue measure) is a collection of point masses.
Definition and properties
The simplest example of a discrete measure on the real line is the Dirac delta function One has and
More generally, if is a (possibly finite) sequence of real numbers, is a sequence of numbers in of the same length, one can consider the Dirac measures defined by
for any Lebesgue measurable set Then, the measure
is a discrete measure. In fact, one may prove that any discrete measure on the real line has this form for appropriately chosen sequences and
One may extend the notion of discrete measures to more general measure spaces. Given a measure space and two measures and on it, is said to be discrete in respect to if there exists an at most countable subset of such that
- All singletons with in are measurable (which implies that any subset of is measurable)
Notice that the first two requirements are always satisfied for an at most countable subset of the real line if is the Lebesgue measure, so they were not necessary in the first definition above.
As in the case of measures on the real line, a measure on is discrete in respect to another measure on the same space if and only if has the form
where the singletons are in and their measure is 0.
One can also define the concept of discreteness for signed measures. Then, instead of conditions 2 and 3 above one should ask that be zero on all measurable subsets of and be zero on measurable subsets of
- Kurbatov, V. G. (1999). Functional differential operators and equations. Kluwer Academic Publishers. ISBN 0-7923-5624-1.