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 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

Template:See also Given two (positive) σ-finite measures $\mu$ and $\nu$ on a measurable space $(X,\mathrm{\Sigma })$. Then $\mu$ is said to be discrete with respect to $\nu$ if there exists an at most countable subset $S\subset X$ in $\mathrm{\Sigma }$ 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 $(X,\mathrm{\Sigma })$ is discrete (with respect to $\nu$) if and only if $\mu$ has the form

$\mu =\sum _{i=1}^{\mathrm{\infty }}{a}_i{\delta }_{s_i}$

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

${\delta }_{s_i}(X)=\{\begin{array}{cc}1& \text{ if }{s}_i\in X\\ 0& \text{ if }{s}_i\in ̸X\end{array}$

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 $X\mathrm{\setminus }S.$Template:Clarify

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

$s_1,s_2,\dots$

such that

$\mu (ℝ\mathrm{\setminus }\{s_1,s_2,\dots \})=0.$

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 $\delta (ℝ\mathrm{\setminus }\{0\})=0$ 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 $[0,\mathrm{\infty }]$ of the same length.

See also

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References

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External links

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