information:extraction, structural:summarization, quality:correction
As a measure

One way to rigorously capture the notion of the Dirac delta function is to define a measure, called Dirac measure, which accepts a subset A of the real line R as an argument, and returns δ (A) = 1 if 0 ∈ A, and δ (A) = 0 otherwise. If the delta function is conceptualized as modeling an idealized point mass at 0, then δ (A) represents the mass contained in the set A. One may then define the integral against 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 δ satisfies

∫ − ∞ ∞ f (x) δ (d x) = f (0) {\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (dx)=f(0)}

for all continuous compactly supported functions f. The measure 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

∫ − ∞ ∞ f (x) δ (x) d x = f (0) {\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (x)\,dx=f(0)}

holds. As a result, the latter notation is a convenient abuse of notation, and not a standard (Riemann or Lebesgue) integral.

As a probability measure on R, the delta measure is characterized by its cumulative distribution function, which is the unit step function.

H (x) = { 1 if x ≥ 0 0 if x < 0. {\displaystyle H(x)={\begin{cases}1&{\text{if }}x\geq 0\\0&{\text{if }}x<0.\end{cases}}}

This means that H (x) is the integral of the cumulative indicator function 1 with respect to the measure δ; to wit,

H (x) = ∫ R 1 (− ∞, x ] (t) δ (d t) = δ ((− ∞, x ]), {\displaystyle H(x)=\int _{\mathbf {R} }\mathbf {1} _{(-\infty,x]}(t)\,\delta (dt)=\delta \!\left((-\infty,x]\right),}

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:

∫ − ∞ ∞ f (x) δ (d x) = ∫ − ∞ ∞ f (x) d H (x). {\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (dx)=\int _{-\infty }^{\infty }f(x)\,dH(x).}

All higher moments of δ are zero. In particular, characteristic function and moment generating function are both equal to one.