Editing Dirac delta function (section)

===As a measure===
One way to rigorously capture the notion of the Dirac delta function is to define a [[Measure (mathematics)|measure]], called [[Dirac measure]], which accepts a subset {{mvar|A}} of the real line {{math|'''R'''}} as an argument, and returns {{math|1=''δ''(''A'') = 1}} if {{math|0 ∈ ''A''}}, and {{math|1=''δ''(''A'') = 0}} otherwise.<ref name="Rudin 1966 loc=§1.20">{{harvnb|Rudin|1966|loc=§1.20}}</ref>  If the delta function is conceptualized as modeling an idealized point mass at 0, then {{math|''δ''(''A'')}} represents the mass contained in the set {{mvar|A}}.  One may then define the integral against {{mvar|δ}} as the integral of a function against this mass distribution.  Formally, the [[Lebesgue integral]] provides the necessary analytic device. The Lebesgue integral with respect to the measure {{mvar|δ}} satisfies

<math display="block">\int_{-\infty}^\infty f(x) \, \delta(dx) =  f(0)</math>

for all continuous compactly supported functions {{mvar|f}}. The measure {{mvar|δ}} is not [[absolutely continuous]] with respect to the [[Lebesgue measure]]—in fact, it is a [[singular measure]]. Consequently, the delta measure has no [[Radon–Nikodym derivative]] (with respect to Lebesgue measure)—no true function for which the property

<math display="block">\int_{-\infty}^\infty f(x)\, \delta(x)\, dx = f(0)</math>

holds.{{sfn|Hewitt|Stromberg|1963|loc=§19.61}} As a result, the latter notation is a convenient [[abuse of notation]], and not a standard ([[Riemann integral|Riemann]] or [[Lebesgue integral|Lebesgue]]) integral.

As a [[probability measure]] on {{math|'''R'''}}, the delta measure is characterized by its [[cumulative distribution function]], which is the [[unit step function]].<ref>{{harvnb|Driggers|2003|p=2321}}  See also {{harvnb|Bracewell|1986|loc=Chapter 5}} for a different interpretation.  Other conventions for the assigning the value of the Heaviside function at zero exist, and some of these are not consistent with what follows.</ref>

<math display="block">H(x) = 
\begin{cases}
1 & \text{if } x\ge 0\\
0 & \text{if } x < 0.
\end{cases}</math>

This means that {{math|''H''(''x'')}} is the integral of the cumulative [[indicator function]] {{math|'''1'''<sub>(−∞, ''x'']</sub>}} with respect to the measure {{mvar|δ}}; to wit,

<math display="block">H(x) = \int_{\mathbf{R}}\mathbf{1}_{(-\infty,x]}(t)\,\delta(dt) = \delta\!\left((-\infty,x]\right),</math>

the latter being the measure of this interval. Thus in particular the integration of the delta function against a continuous function can be properly understood as a [[Riemann–Stieltjes integral]]:{{sfn|Hewitt|Stromberg|1963|loc=§9.19}}

<math display="block">\int_{-\infty}^\infty f(x)\,\delta(dx) = \int_{-\infty}^\infty f(x) \,dH(x).</math>

All higher [[moment (mathematics)|moments]] of {{mvar|δ}} are zero.  In particular, [[characteristic function (probability theory)|characteristic function]] and [[moment generating function]] are both equal to one.