Editing Dirac delta function (section)

===Generalizations===
The delta function can be defined in {{mvar|n}}-dimensional [[Euclidean space]] {{math|'''R'''<sup>''n''</sup>}} as the measure such that

<math display="block">\int_{\mathbf{R}^n} f(\mathbf{x})\,\delta(d\mathbf{x}) = f(\mathbf{0})</math>

for every compactly supported continuous function {{mvar|f}}.  As a measure, the {{mvar|n}}-dimensional delta function is the [[product measure]] of the 1-dimensional delta functions in each variable separately. Thus, formally, with {{math|1='''x''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>)}}, one has{{sfn|Bracewell|1986|loc=Chapter 5}}

{{NumBlk2|:|<math>\delta(\mathbf{x}) = \delta(x_1)\,\delta(x_2)\cdots\delta(x_n).</math>|2}}

The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case.{{sfn|Hörmander|1983|loc=§3.1}}  However, despite widespread use in engineering contexts, ({{EquationNote|2}}) should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances.{{sfn|Strichartz|1994|loc=§2.3}}{{sfn|Hörmander|1983|loc=§8.2}}

The notion of a '''[[Dirac measure]]''' makes sense on any set.{{sfn|Rudin |1966 |loc=§1.20}} Thus if {{mvar|X}} is a set, {{math|''x''<sub>0</sub> ∈ ''X''}} is a marked point, and {{math|Σ}} is any [[sigma algebra]] of subsets of {{mvar|X}}, then the measure defined on sets {{math|''A'' ∈ Σ}} by

<math display="block">\delta_{x_0}(A)=\begin{cases}
1 &\text{if }x_0\in A\\
0 &\text{if }x_0\notin A
\end{cases}</math>

is the delta measure or unit mass concentrated at {{math|''x''<sub>0</sub>}}.

Another common generalization of the delta function is to a [[differentiable manifold]] where most of its properties as a distribution can also be exploited because of the [[differentiable structure]].  The delta function on a manifold {{mvar|M}} centered at the point {{math|''x''<sub>0</sub> ∈ ''M''}} is defined as the following distribution:

{{NumBlk2|:|<math>\delta_{x_0}[\varphi] = \varphi(x_0)</math>|3}}

for all compactly supported smooth real-valued functions {{mvar|φ}} on {{mvar|M}}.{{sfn|Dieudonné|1972|loc=§17.3.3}} A common special case of this construction is a case in which {{mvar|M}} is an [[open set]] in the Euclidean space {{math|'''R'''<sup>''n''</sup>}}.

On a [[locally compact Hausdorff space]] {{mvar|X}}, the Dirac delta measure concentrated at a point {{mvar|x}} is the [[Radon measure]] associated with the Daniell integral ({{EquationNote|3}}) on compactly supported continuous functions {{mvar|φ}}.<ref>{{Cite book|last1=Krantz|first1=Steven G.|url={{google books |plainurl=y |id=X_BKmVphIcsC&q }}|title=Geometric Integration Theory|last2=Parks|first2=Harold R.|date=2008-12-15|publisher=Springer Science & Business Media|isbn=978-0-8176-4679-0|language=en}}</ref> At this level of generality, calculus as such is no longer possible, however a variety of techniques from abstract analysis are available. For instance, the mapping <math>x_0\mapsto \delta_{x_0}</math> is a continuous embedding of {{mvar|X}} into the space of finite Radon measures on {{mvar|X}}, equipped with its [[vague topology]]. Moreover, the [[convex hull]] of the image of {{mvar|X}} under this embedding is [[dense set|dense]] in the space of probability measures on {{mvar|X}}.{{sfn|Federer|1969|loc=§2.5.19}}