Bethe formula

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 $[0,\mathrm{\infty }]$ is said to be discrete if there exists a (possibly finite) sequence of numbers

$s_1,s_2,\dots$

such that

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

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

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 $[0,\mathrm{\infty }]$ of the same length, one can consider the Dirac measures ${\delta }_{s_i}$ defined by

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

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

One may extend the notion of discrete measures to more general measure spaces. Given a measure space $(X,\mathrm{\Sigma }),$ and two measures $\mu$ and $\nu$ on it, $\mu$ is said to be discrete in respect to $\nu$ if there exists an at most countable subset $S$ of $X$ 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. $\mu (X\mathrm{\setminus }S)=0.$

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.

As in the case of measures on the real line, a measure $\mu$ on $(X,\mathrm{\Sigma })$ is discrete in respect to another measure $\nu$ on the same space if and only if $\mu$ has the form

$\mu =\sum _i{a}_i{\delta }_{s_i}$

where $S=\{s_1,s_2,\dots \},$ the singletons $\{s_i\}$ are in $\mathrm{\Sigma },$ and their $\nu$ 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 $\nu$ be zero on all measurable subsets of $S$ and $\mu$ be zero on measurable subsets of $X\mathrm{\setminus }S.$

References

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