Discrete measure

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 concentrated on an at most countable set. 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.

## Contents

* 1 Definition and properties
* 2 Example on R
* 3 See also
* 4 References
* 5 External links

## Definition and properties

Given two (positive) σ-finite measures μ and ν on a measurable space (X,Σ). Then μ is said to be discrete with respect to ν if there exists an at most countable subset S⊂X in Σ such that

1. All singletons {s} with s∈S are measurable (which implies that any subset of S is measurable)
2. ν(S)=0
3. μ(X∖S)=0.

A measure μ on (X,Σ) is discrete (with respect to ν) if and only if μ has the form

μ=∑i=1∞aiδsi

with ai∈ℝ>0 and Dirac measures δsi on the set S={si}i∈ℕ defined as

δsi(X)={1 if si∈X0 if si∉X

for all i∈ℕ.

One can also define the concept of discreteness for signed measures. Then, instead of conditions 2 and 3 above one should ask that ν be zero on all measurable subsets of S and μ be zero on measurable subsets of X∖S.[clarification needed]

## Example on R

A measure μ defined on the Lebesgue measurable sets of the real line with values in [0,∞] is said to be discrete if there exists a (possibly finite) sequence of numbers

s1,s2,…

such that

μ(ℝ∖{s1,s2,…})=0.

Notice that the first two requirements in the previous section are always satisfied for an at most countable subset of the real line if ν is the Lebesgue measure.

The simplest example of a discrete measure on the real line is the Dirac delta function δ. One has δ(ℝ∖{0})=0 and δ({0})=1.

More generally, one may prove that any discrete measure on the real line has the form

μ=∑iaiδsi

for an appropriately chosen (possibly finite) sequence s1,s2,… of real numbers and a sequence a1,a2,… of numbers in [0,∞] of the same length.

## See also

* Isolated point - Point of a subset S around which there are no other points of S
* Lebesgue's decomposition theorem - Theorem
* Singleton (mathematics) - Set with exactly one element
* Singular measure - Probability distribution in measure theory

## References

* "Why must a discrete atomic measure admit a decomposition into Dirac measures? Moreover, what is "an atomic class"?". Feb 24, 2022. https://math.stackexchange.com/a/4391020.
* Kurbatov, V. G. (1999). Functional differential operators and equations. Kluwer Academic Publishers. ISBN 0-7923-5624-1.

## External links

* Hazewinkel, Michiel, ed. (2001), "Discrete measure", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=d/d033090

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