# Discrete measure

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. 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. 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.

## 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 measurable 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 $X\backslash S.$