Bethe formula

2013-10-20T23:06:57Z

98.195.217.66: /* The formula */

In [[mathematics]], more precisely in [[measure theory]], a [[measure (mathematics)|measure]] on the [[real line]] is called a '''discrete measure''' (in respect to the [[Lebesgue measure]]) if its [[support (measure theory)|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 <math>\mu</math> defined on the [[Lebesgue measure|Lebesgue measurable set]]s of the real line with values in <math>[0, \infty]</math> is said to be '''discrete''' if there exists a (possibly finite) [[sequence]] of numbers

: <math>s_1, s_2, \dots \,</math>

such that
: <math>\mu(\mathbb R\backslash\{s_1, s_2, \dots\})=0.</math>

The simplest example of a discrete measure on the real line is the [[Dirac delta function]] <math>\delta.</math> One has <math>\delta(\mathbb R\backslash\{0\})=0</math> and <math>\delta(\{0\})=1.</math>

More generally, if <math>s_1, s_2, \dots</math> is a (possibly finite) sequence of real numbers, <math>a_1, a_2, \dots</math> is a sequence of numbers in <math>[0, \infty]</math> of the same length, one can consider the [[Dirac measure]]s <math>\delta_{s_i}</math> defined by

: <math>\delta_{s_i}(X) =
\begin{cases}
1 & \mbox { if } s_i \in X\\
0 & \mbox { if } s_i \not\in X\\
\end{cases}
</math>
for any Lebesgue measurable set <math>X.</math> Then, the measure

: <math>\mu = \sum_{i} a_i \delta_{s_i}</math>

is a discrete measure. In fact, one may prove that any discrete measure on the real line has this form for appropriately chosen sequences <math>s_1, s_2, \dots</math> and <math>a_1, a_2, \dots</math>

==Extensions==

One may extend the notion of discrete measures to more general [[measure space]]s. Given a measure space <math>(X, \Sigma),</math> and two measures <math>\mu</math> and <math>\nu</math> on it, <math>\mu</math> is said to be '''discrete''' in respect to <math>\nu</math> if there exists an at most countable subset <math>S</math> of <math>X</math> such that
# All singletons <math>\{s\}</math> with <math>s</math> in <math>S</math> are measurable (which implies that any subset of <math>S</math> is measurable)
# <math>\nu(S)=0\,</math>
# <math>\mu(X\backslash S)=0.\,</math>
Notice that the first two requirements are always satisfied for an at most countable subset of the real line if <math>\nu</math> 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 <math>\mu</math> on <math>(X, \Sigma)</math> is discrete in respect to another measure <math>\nu</math> on the same space if and only if <math>\mu</math> has the form

: <math>\mu = \sum_{i} a_i \delta_{s_i}</math>

where <math>S=\{s_1, s_2, \dots\},</math> the singletons <math>\{s_i\}</math> are in <math>\Sigma,</math> and their <math>\nu</math> measure is 0.

One can also define the concept of discreteness for [[signed measure]]s. Then, instead of conditions 2 and 3 above one should ask that <math>\nu</math> be zero on all measurable subsets of <math>S</math> and <math>\mu</math> be zero on measurable subsets of <math>X\backslash S.</math>

==References==

* {{cite book
| last = Kurbatov
| first = V. G.
| title = Functional differential operators and equations
| publisher = Kluwer Academic Publishers
| year = 1999
| pages =
| isbn = 0-7923-5624-1
}}

==External links==
* {{springer|id=d/d033090|title=Discrete measure|author=A.P. Terekhin}}

[[Category:Measures (measure theory)]]

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Bethe formula