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 propertiesEdit

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

 

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

More generally, if   is a (possibly finite) sequence of real numbers,   is a sequence of numbers in   of the same length, one can consider the Dirac measures   defined by

 

for any Lebesgue measurable set   Then, the measure

 

is a discrete measure. In fact, one may prove that any discrete measure on the real line has this form for appropriately chosen sequences   and  

ExtensionsEdit

One may extend the notion of discrete measures to more general measure spaces. Given a measurable space   and two measures   and   on it,   is said to be discrete in respect to   if there exists an at most countable subset   of   such that

  1. All singletons   with   in   are measurable (which implies that any subset of   is measurable)
  2.  
  3.  

Notice that the first two requirements are always satisfied for an at most countable subset of the real line if   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   on   is discrete in respect to another measure   on the same space if and only if   has the form

 

where   the singletons   are in   and their   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   be zero on all measurable subsets of   and   be zero on measurable subsets of  

ReferencesEdit

  • Kurbatov, V. G. (1999). Functional differential operators and equations. Kluwer Academic Publishers. ISBN 0-7923-5624-1.

External linksEdit