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.

Definition and properties

Given two (positive) σ-finite measures [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \nu }[/math] on a measurable space [math]\displaystyle{ (X, \Sigma) }[/math]. Then [math]\displaystyle{ \mu }[/math] is said to be discrete with respect to [math]\displaystyle{ \nu }[/math] if there exists an at most countable subset [math]\displaystyle{ S \subset X }[/math] in [math]\displaystyle{ \Sigma }[/math] such that

1. All singletons [math]\displaystyle{ \{s\} }[/math] with [math]\displaystyle{ s \in S }[/math] are measurable (which implies that any subset of [math]\displaystyle{ S }[/math] is measurable)
2. [math]\displaystyle{ \nu(S)=0\, }[/math]
3. [math]\displaystyle{ \mu(X\setminus S)=0.\, }[/math]

A measure [math]\displaystyle{ \mu }[/math] on [math]\displaystyle{ (X, \Sigma) }[/math] is discrete (with respect to [math]\displaystyle{ \nu }[/math]) if and only if [math]\displaystyle{ \mu }[/math] has the form

[math]\displaystyle{ \mu = \sum_{i=1}^{\infty} a_i \delta_{s_i} }[/math]

with [math]\displaystyle{ a_i \in \mathbb{R}_{\gt 0} }[/math] and Dirac measures [math]\displaystyle{ \delta_{s_i} }[/math] on the set [math]\displaystyle{ S=\{s_i\}_{i\in\mathbb{N}} }[/math] defined as

[math]\displaystyle{ \delta_{s_i}(X) = \begin{cases} 1 & \mbox { if } s_i \in X\\ 0 & \mbox { if } s_i \not\in X\\ \end{cases} }[/math]

for all [math]\displaystyle{ i\in\mathbb{N} }[/math].

One can also define the concept of discreteness for signed measures. Then, instead of conditions 2 and 3 above one should ask that [math]\displaystyle{ \nu }[/math] be zero on all measurable subsets of [math]\displaystyle{ S }[/math] and [math]\displaystyle{ \mu }[/math] be zero on measurable subsets of [math]\displaystyle{ X\backslash S. }[/math]

Example on R

A measure [math]\displaystyle{ \mu }[/math] defined on the Lebesgue measurable sets of the real line with values in [math]\displaystyle{ [0, \infty] }[/math] is said to be discrete if there exists a (possibly finite) sequence of numbers

[math]\displaystyle{ s_1, s_2, \dots \, }[/math]

such that

[math]\displaystyle{ \mu(\mathbb R\backslash\{s_1, s_2, \dots\})=0. }[/math]

Notice that the first two requirements in the previous section are always satisfied for an at most countable subset of the real line if [math]\displaystyle{ \nu }[/math] is the Lebesgue measure.

The simplest example of a discrete measure on the real line is the Dirac delta function [math]\displaystyle{ \delta. }[/math] One has [math]\displaystyle{ \delta(\mathbb R\backslash\{0\})=0 }[/math] and [math]\displaystyle{ \delta(\{0\})=1. }[/math]

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

[math]\displaystyle{ \mu = \sum_{i} a_i \delta_{s_i} }[/math]

for an appropriately chosen (possibly finite) sequence [math]\displaystyle{ s_1, s_2, \dots }[/math] of real numbers and a sequence [math]\displaystyle{ a_1, a_2, \dots }[/math] of numbers in [math]\displaystyle{ [0, \infty] }[/math] 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

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