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

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 $X\backslash S.$

Example on 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

• Isolated point – Point of a subset S around which there are no other points of S
• Lebesgue's decomposition theorem
• Singleton (mathematics) – Set with exactly one element
• Singular measure – measure or probability distribution whose support has zero Lebesgue (or other) measurePages displaying wikidata descriptions as a fallback

References

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

External links

• A.P. Terekhin (2001) [1994], "Discrete measure", Encyclopedia of Mathematics, EMS Press