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 at most concentrated on a countable set. Note that 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

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

$\mu(\mathbb R\backslash\{s_1, s_2, \dots\})=0.$

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, if $s_1, s_2, \dots$ is a (possibly finite) sequence of real numbers, $a_1, a_2, \dots$ is a sequence of numbers in $[0, \infty]$ of the same length, one can consider the Dirac measures $\delta_{s_i}$ defined by

$\delta_{s_i}(X) = \begin{cases} 1 & \mbox { if } s_i \in X\\ 0 & \mbox { if } s_i \not\in X\\ \end{cases}$

for any Lebesgue measurable set $X.$ Then, the measure

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

is a discrete measure. In fact, one may prove that any discrete measure on the real line has this form for appropriately chosen sequences $s_1, s_2, \dots$ and $a_1, a_2, \dots$

Extensions

One may extend the notion of discrete measures to more general measure spaces. Given a measure space $(X, \Sigma),$ and two measures $\mu$ and $\nu$ on it, $\mu$ is said to be discrete in respect to $\nu$ if there exists an at most countable subset $S$ of $X$ 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. $\mu(X\backslash S)=0.\,$

Notice that the first two requirements are always satisfied for an at most countable subset of the real line if $\nu$ is the Lebesgue measure, so they were not necessary in the first definition above.

As in the case of measures on the real line, a measure $\mu$ on $(X, \Sigma)$ is discrete in respect to another measure $\nu$ on the same space if and only if $\mu$ has the form

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

where $S=\{s_1, s_2, \dots\},$ the singletons $\{s_i\}$ are in $\Sigma,$ and their $\nu$ measure is 0.

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 $X\backslash S.$

References

• Kurbatov, V. G. (1999). Functional differential operators and equations. Kluwer Academic Publishers. ISBN 0-7923-5624-1.

External links

• A.P. Terekhin (2001), "Discrete measure", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4