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.

1 Definition and properties
2 Extensions
3 References
4 External links

[edit] Definition and properties

More precisely, a measure $μ$ 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 $δ.$ One has $\delta(\mathbb R\backslash\{0\})=0$ and $δ({0}) = 1.$

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 $a_1, a_2, \dots$

[edit] Extensions

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

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

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

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

where $S=\{s_1, s_2, \dots\},$ the singletons ${s i}$ are in $Σ,$ and their $ν$ 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 $ν$ be zero on all measurable subsets of $S$ and $μ$ be zero on measurable subsets of $X\backslash S.$