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. 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 measure, Lebesgue measurable sets of the real line with values in , \infty/math> 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\)=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$ and $\delta(\)=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 , \infty/math> of the same length, one can consider the Dirac measures $\delta_$ defined by

: $\delta_(X) = \begin 1 & \mbox s_i \in X\\ 0 & \mbox s_i \not\in X\\ \end$
for any Lebesgue measurable set $X.$ Then, the measure

: $\mu = \sum_ a_i \delta_$

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
# All singletons $\$ with $s \in S$ are measurable (which implies that any subset of $S$ is measurable)
# $\nu(S)=0\,$
# $\mu(X\backslash S)=0.\,$
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

: $\mu = \sum_ a_i \delta_$

where $S=\,$ the singletons $\$ 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

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External links

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{{Measure theory

Measures (measure theory)