Discrete measure

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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 its support is at most a 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.

Definition and properties[edit]

A measure \mu defined on the Lebesgue measurable sets of the real line with values in [0, \infty] 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\{s_1, s_2, \dots\})=0.

The simplest example of a discrete measure on the real line is the Dirac delta function \delta. One has \delta(\mathbb R\backslash\{0\})=0 and \delta(\{0\})=1.

More generally, if s_1, s_2, \dots is a (possibly finite) sequence of real numbers, a_1, a_2, \dots is a sequence of numbers in [0, \infty] of the same length, one can consider the Dirac measures \delta_{s_i} defined by

\delta_{s_i}(X) = 
\begin{cases} 
1 & \mbox { if } s_i \in X\\ 
0 & \mbox { if } s_i \not\in X\\ 
\end{cases}

for any Lebesgue measurable set X. Then, the measure

\mu = \sum_{i} a_i \delta_{s_i}

is a discrete measure. In fact, one may prove that any discrete measure on the real line has this form for appropriately chosen sequences s_1, s_2, \dots and a_1, a_2, \dots

Extensions[edit]

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

  1. All singletons \{s\} with s in S are measurable (which implies that any subset of S is measurable)
  2. \nu(S)=0\,
  3. \mu(X\backslash S)=0.\,

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

\mu = \sum_{i} a_i \delta_{s_i}

where S=\{s_1, s_2, \dots\}, the singletons \{s_i\} are in \Sigma, and their \nu 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 \nu be zero on all measurable subsets of S and \mu be zero on measurable subsets of X\backslash S.

References[edit]

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

External links[edit]