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.

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 properties

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Given two (positive) σ-finite measures   and   on a measurable space  . Then   is said to be discrete with respect to   if there exists an at most countable subset   in   such that

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

A measure   on   is discrete (with respect to  ) if and only if   has the form

 

with   and Dirac measures   on the set   defined as

 

for all  .

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  [clarification needed]

Example on R

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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   One has   and  

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   of the same length.

See also

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References

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  • "Why must a discrete atomic measure admit a decomposition into Dirac measures? Moreover, what is "an atomic class"?". math.stackexchange.com. Feb 24, 2022.
  • Kurbatov, V. G. (1999). Functional differential operators and equations. Kluwer Academic Publishers. ISBN 0-7923-5624-1.
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