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.

Given two (positive) σ-finite measures

μ

on a measurable space

μ

is said to be discrete with respect to

if there exists an at most countable subset

is discrete (with respect to

has the form with

and Dirac measures

One can also define the concept of discreteness for signed measures.

be zero on all measurable subsets of

be zero on measurable subsets of

[clarification needed] A measure

defined on the Lebesgue measurable sets of the real line with values in

is said to be discrete if there exists a (possibly finite) sequence of numbers such that Notice that the first two requirements in the previous section are always satisfied for an at most countable subset of the real line if

is the Lebesgue measure.

The simplest example of a discrete measure on the real line is the Dirac delta function

More generally, one may prove that any discrete measure on the real line has the form for an appropriately chosen (possibly finite) sequence

of real numbers and a sequence

of numbers in