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 ${\displaystyle \mu }$ and ${\displaystyle \nu }$ on a measurable space ${\displaystyle (X,\Sigma )}$. Then ${\displaystyle \mu }$ is said to be discrete with respect to ${\displaystyle \nu }$ if there exists an at most countable subset ${\displaystyle S\subset X}$ in ${\displaystyle \Sigma }$ such that

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

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

${\displaystyle \mu =\sum _{i=1}^{\infty }a_{i}\delta _{s_{i}}}$

with ${\displaystyle a_{i}\in \mathbb {R} _{>0}}$ and Dirac measures ${\displaystyle \delta _{s_{i}}}$ on the set ${\displaystyle S=\{s_{i}\}_{i\in \mathbb {N} }}$ defined as

${\displaystyle \delta _{s_{i}}(X)={\begin{cases}1&{\mbox{ if }}s_{i}\in X\\0&{\mbox{ if }}s_{i}\not \in X\\\end{cases}}}$

for all ${\displaystyle i\in \mathbb {N} }$.

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

Example on R

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

${\displaystyle s_{1},s_{2},\dots \,}$

such that

${\displaystyle \mu (\mathbb {R} \backslash \{s_{1},s_{2},\dots \})=0.}$

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

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

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

${\displaystyle \mu =\sum _{i}a_{i}\delta _{s_{i}}}$

for an appropriately chosen (possibly finite) sequence ${\displaystyle s_{1},s_{2},\dots }$ of real numbers and a sequence ${\displaystyle a_{1},a_{2},\dots }$ of numbers in ${\displaystyle [0,\infty ]}$ 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