Discrete measure

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Schematic representation of the Dirac measure by a line surmounted by an arrow. The Dirac measure is a discrete measure whose support is the point 0. The Dirac measure of any set containing 0 is 1, and the measure of any set not containing 0 is 0.

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 it is at most concentrated on 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[edit]

A measure defined on the Lebesgue measurable sets of the real line with values in is said to be discrete if there exists a (possibly finite) sequence of numbers

such that

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

Extensions[edit]

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

  1. 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

References[edit]

  • Kurbatov, V. G. (1999). Functional differential operators and equations. Kluwer Academic Publishers. ISBN 0-7923-5624-1. 

External links[edit]