# Discrete measure

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff) 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.

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

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

$\mu (\mathbb {R} \backslash \{s_{1},s_{2},\dots \})=0.$ $\delta _{s_{i}}(X)={\begin{cases}1&{\mbox{ if }}s_{i}\in X\\0&{\mbox{ if }}s_{i}\not \in X\\\end{cases}}$ 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

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.

$\mu =\sum _{i}a_{i}\delta _{s_{i}}$ 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 $X\backslash S.$ 