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.

Definition and properties
Given two (positive) σ-finite measures $$\mu$$ 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
 * 1) All singletons $$\{s\}$$ with $$s \in S$$ are measurable  (which implies that any subset of $$S$$ is measurable)
 * 2) $$\nu(S)=0\,$$
 * 3) $$\mu(X\setminus S)=0.\,$$

A measure $$\mu$$ on $$(X, \Sigma)$$ is discrete (with respect to $$\nu$$) if and only if $$\mu$$ has the form
 * $$\mu = \sum_{i=1}^{\infty} a_i \delta_{s_i}$$

with $$ a_i \in \mathbb{R}_{>0}$$ and Dirac measures $$\delta_{s_i}$$ on the set $$S=\{s_i\}_{i\in\mathbb{N}}$$ defined as
 * $$\delta_{s_i}(X) =

\begin{cases} 1 & \mbox { if } s_i \in X\\ 0 & \mbox { if } s_i \not\in X\\ \end{cases} $$ for all $$i\in\mathbb{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 $R$
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
 * $$\mu(\mathbb R\backslash\{s_1, s_2, \dots\})=0.$$

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.

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, 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 $$s_1, s_2, \dots$$ of real numbers and a sequence $$a_1, a_2, \dots$$ of numbers in $$[0, \infty]$$ of the same length.