Файл:Dirac distribution PDF.svgSchematic 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.
Шаблон:See also
Given two (positive) σ-finite measures <math>\mu</math> and <math>\nu</math> on a measurable space <math>(X, \Sigma)</math>. Then <math>\mu</math> is said to be discrete with respect to <math>\nu</math> if there exists an at most countable subset <math>S \subset X</math> in <math>\Sigma</math> such that
All singletons <math>\{s\}</math> with <math>s \in S</math> are measurable (which implies that any subset of <math>S</math> is measurable)
<math>\nu(S)=0\,</math>
<math>\mu(X\setminus S)=0.\,</math>
A measure <math>\mu</math> on <math>(X, \Sigma)</math> is discrete (with respect to <math>\nu</math>) if and only if <math>\mu</math> has the form
with <math> a_i \in \mathbb{R}_{>0}</math> and Dirac measures <math>\delta_{s_i}</math> on the set <math>S=\{s_i\}_{i\in\mathbb{N}}</math> defined as
<math>\delta_{s_i}(X) =
\begin{cases}
1 & \mbox { if } s_i \in X\\
0 & \mbox { if } s_i \not\in X\\
\end{cases}
</math>
for all <math>i\in\mathbb{N}</math>.
One can also define the concept of discreteness for signed measures. Then, instead of conditions 2 and 3 above one should ask that <math>\nu</math> be zero on all measurable subsets of <math>S</math> and <math>\mu</math> be zero on measurable subsets of <math>X\backslash S.</math>Шаблон:Clarify
Example on R
A measure <math>\mu</math> defined on the Lebesgue measurable sets of the real line with values in <math>[0, \infty]</math> is said to be discrete if there exists a (possibly finite) sequence of numbers
Notice that the first two requirements in the previous section are always satisfied for an at most countable subset of the real line if <math>\nu</math> is the Lebesgue measure.
The simplest example of a discrete measure on the real line is the Dirac delta function <math>\delta.</math> One has <math>\delta(\mathbb R\backslash\{0\})=0</math> and <math>\delta(\{0\})=1.</math>
More generally, one may prove that any discrete measure on the real line has the form
<math>\mu = \sum_{i} a_i \delta_{s_i}</math>
for an appropriately chosen (possibly finite) sequence <math>s_1, s_2, \dots</math> of real numbers and a sequence <math>a_1, a_2, \dots</math> of numbers in <math>[0, \infty]</math> of the same length.
