Discrete measure

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 ${\displaystyle \mu }$ defined on the Lebesgue measurable sets of the real line with values in ${\displaystyle [0,\infty ]}$ is said to be discrete if there exists a (possibly finite) sequence of numbers

${\displaystyle s_{1},s_{2},\dots \,}$

such that

${\displaystyle \mu (\mathbb {R} \backslash \{s_{1},s_{2},\dots \})=0.}$

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

More generally, if ${\displaystyle s_{1},s_{2},\dots }$ is a (possibly finite) sequence of real numbers, ${\displaystyle a_{1},a_{2},\dots }$ is a sequence of numbers in ${\displaystyle [0,\infty ]}$ of the same length, one can consider the Dirac measures ${\displaystyle \delta _{s_{i}}}$ defined by

${\displaystyle \delta _{s_{i}}(X)={\begin{cases}1&{\mbox{ if }}s_{i}\in X\\0&{\mbox{ if }}s_{i}\not \in X\\\end{cases}}}$

for any Lebesgue measurable set ${\displaystyle X.}$ Then, the measure

${\displaystyle \mu =\sum _{i}a_{i}\delta _{s_{i}}}$

is a discrete measure. In fact, one may prove that any discrete measure on the real line has this form for appropriately chosen sequences ${\displaystyle s_{1},s_{2},\dots }$ and ${\displaystyle a_{1},a_{2},\dots }$

Extensions

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

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

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

${\displaystyle \mu =\sum _{i}a_{i}\delta _{s_{i}}}$

One can also define the concept of discreteness for signed measures. Then, instead of conditions 2 and 3 above one should ask that ${\displaystyle \nu }$ be zero on all measurable subsets of ${\displaystyle S}$ and ${\displaystyle \mu }$ be zero on measurable subsets of ${\displaystyle X\backslash S.}$

References

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