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Discrete measure

File:Dirac distribution PDF.svg 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

{{See also|Atom (measure theory)}}

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

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

A measure \mu on (X, \Sigma) is discrete (with respect to \nu) if and only if \mu has the form

:\mu = \sum_{i=1}^{\infty} a_i \delta_{s_i}

with a_i \in \mathbb{R}_{>0} and Dirac measures \delta_{s_i} on the set S=\{s_i\}_{i\in\mathbb{N}} defined as

:\delta_{s_i}(X) =

\begin{cases}

1 & \mbox { if } s_i \in X\\

0 & \mbox { if } s_i \not\in X\\

\end{cases}

for all i\in\mathbb{N}.

One can also define the concept of discreteness for signed measures. Then, instead of conditions 2 and 3 above one should ask that \nu be zero on all measurable subsets of S and \mu be zero on measurable subsets of X\backslash S.{{clarify|reason=How is that different form the current conditions 2 and 3?|date=November 2023}}

Example on {{math|R}}

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

: s_1, s_2, \dots \,

such that

: \mu(\mathbb R\backslash\{s_1, s_2, \dots\})=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 \nu is the Lebesgue measure.

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

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

:\mu = \sum_{i} a_i \delta_{s_i}

for an appropriately chosen (possibly finite) sequence s_1, s_2, \dots of real numbers and a sequence a_1, a_2, \dots of numbers in [0, \infty] of the same length.

See also

  • {{annotated link|Isolated point}}
  • {{annotated link|Lebesgue's decomposition theorem}}
  • {{annotated link|Singleton (mathematics)}}
  • {{annotated link|Singular measure}}

References

  • {{Cite web|url=https://math.stackexchange.com/a/4391020|title = Why must a discrete atomic measure admit a decomposition into Dirac measures? Moreover, what is "an atomic class"?| date = Feb 24, 2022|website=math.stackexchange.com}}
  • {{cite book

| last = Kurbatov

| first = V. G.

| title = Functional differential operators and equations

| publisher = Kluwer Academic Publishers

| year = 1999

| pages =

| isbn = 0-7923-5624-1

}}

External links

  • {{springer|id=d/d033090|title=Discrete measure|author=A.P. Terekhin}}

{{Measure theory}}

Category:Measures (measure theory)