Discrete measure explained

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

\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\{s\} 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 ^{inf ty} _{i =1} a _{i} \delta _{s _{i}} with a _{i} \in R _{>0} and Dirac measures \delta _{s _{i}} on the set S = \{s _{i\}} _{i\inN} defined as\delta _{s _{i}} (X) = \begin{cases} 1 & if s _{i} \in X\ 0 & if s _{i} \not\in X\ \end{cases} for all i\in N .

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 [ 0 , inf ty] is said to be discrete if there exists a (possibly finite) sequence of numberss _{1,} s _{2,} ...

such that

\mu( R\backslash\{s _{1,} s _{2,} ... \}) =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.\delta. One has \delta( R\backslash\{0\})=0 and \delta(\{ 0 \}) =1 .

More generally, one may prove that any discrete measure on the real line has the form

\mu = \sum _{i} a _{i} \delta _{s _{i}} for an appropriately chosen (possibly finite) sequence s _{1,} s _{2,} ... of real numbers and a sequence a _{1,} a _{2,} ... of numbers in [ 0 , inf ty] of the same length.

References

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