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

Given two (positive) σ-finite measures $\mu$ and $\nu$ on a measurable space $(X, \Sigma)$. Then $\mu$ is said to be discrete with respect to $\nu$ if there exists an at most countable subset $S \subset X$ in $\Sigma$ such that
# All singletons $\$ with $s \in S$ are measurable (which implies that any subset of $S$ is measurable)
# $\nu(S)=0\,$
# $\mu(X\setminus S)=0.\,$

A measure $\mu$ on $(X, \Sigma)$ is discrete (with respect to $\nu$) if and only if $\mu$ has the form
:$\mu = \sum_^ a_i \delta_$
with $a_i \in \mathbb_$ and Dirac measures $\delta_$ on the set $S=\_$ defined as
:$\delta_(X) = \begin 1 & \mbox s_i \in X\\ 0 & \mbox s_i \not\in X\\ \end$
for all $i\in\mathbb$.

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 $X\backslash S.$

Example on

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

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, one may prove that any discrete measure on the real line has the form
:$\mu = \sum_ a_i \delta_$
for an appropriately chosen (possibly finite) sequence $s_1, s_2, \dots$ of real numbers and a sequence $a_1, a_2, \dots$ of numbers in , \infty/math> of the same length.

See also

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References

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External links

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{{Measure theory

Measures (measure theory)