Editing Dirac measure

{{Short description|Measure that is 1 if and only if a specified element is in the set}}
[[Image:Hasse diagram of powerset of 3.svg|right|thumb|250px|A diagram showing all possible subsets of a 3-point set {{math|{''x'',''y'',''z''}}}. The Dirac measure {{math|''δ<sub>x</sub>''}} assigns a size of 1 to all sets in the upper-left half of the diagram and 0 to all sets in the lower-right half.]]

In [[mathematics]], a '''Dirac measure''' assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the [[Dirac delta function]], an important tool in physics and other technical fields.

==Definition==
A '''Dirac measure''' is a [[measure (mathematics)|measure]] {{math|''δ''<sub>''x''</sub>}} on a set {{math|''X''}} (with any [[sigma algebra|{{math|''σ''}}-algebra]] of [[subset]]s of {{math|''X''}}) defined for a given {{math|''x'' ∈ ''X''}} and any [[measurable set|(measurable) set]] {{math|''A'' ⊆ ''X''}} by
:<math>\delta_x (A) = 1_A(x)= \begin{cases} 0, & x \not \in A; \\ 1, & x \in A. \end{cases}</math>
where {{math|1<sub>''A''</sub>}} is the [[indicator function]] of {{math|''A''}}.

The Dirac measure is a [[probability measure]], and in terms of probability it represents the [[almost sure]] outcome {{math|''x''}} in the [[sample space]] {{math|''X''}}. We can also say that the measure is a single [[Atom (measure theory)|atom]] at {{math|''x''}}; however, treating the Dirac measure as an atomic measure is not correct when we consider the sequential definition of Dirac delta, as the limit of a [[delta sequence]]{{Dubious|date=January 2022}}. The Dirac measures are the [[extreme point]]s of the convex set of probability measures on {{math|''X''}}.

The name is a back-formation from the [[Dirac delta function]]; considered as a [[Distribution (mathematics)|Schwartz distribution]], for example on the [[real line]], measures can be taken to be a special kind of distribution. The identity
:<math>\int_{X} f(y) \, \mathrm{d} \delta_x (y) = f(x),</math>
which, in the form
:<math>\int_X f(y) \delta_x (y) \, \mathrm{d} y = f(x),</math>
is often taken to be part of the definition of the "delta function", holds as a theorem of [[Lebesgue integration]].

==Properties of the Dirac measure==
Let {{math|''δ''<sub>''x''</sub>}} denote the Dirac measure centred on some fixed point {{math|''x''}} in some [[measurable space]] {{math|(''X'', Σ)}}.
* {{math|''δ''<sub>''x''</sub>}} is a probability measure, and hence a [[finite measure]].

Suppose that {{math|(''X'', ''T'')}} is a [[topological space]] and that {{math|Σ}} is at least as fine as the [[Borel sigma algebra|Borel {{math|''σ''}}-algebra]] {{math|''σ''(''T'')}} on {{math|''X''}}.
* {{math|''δ''<sub>''x''</sub>}} is a [[strictly positive measure]] [[if and only if]] the topology {{math|''T''}} is such that {{math|''x''}} lies within every non-empty open set, e.g. in the case of the [[trivial topology]] {{math|{∅, ''X''}<nowiki/>}}.
* Since {{math|''δ''<sub>''x''</sub>}} is probability measure, it is also a [[locally finite measure]].
* If {{math|''X''}} is a [[Hausdorff space|Hausdorff]] topological space with its Borel {{math|''σ''}}-algebra, then {{math|''δ''<sub>''x''</sub>}} satisfies the condition to be an [[inner regular measure]], since [[Singleton (mathematics)|singleton]] sets such as {{math|{''x''}<nowiki/>}} are always [[compact space|compact]]. Hence, {{math|''δ''<sub>''x''</sub>}} is also a [[Radon measure]].
* Assuming that the topology {{math|''T''}} is fine enough that {{math|{''x''}<nowiki/>}} is closed, which is the case in most applications, the [[Support (measure theory)|support]] of {{math|''δ''<sub>''x''</sub>}} is {{math|{''x''}<nowiki/>}}. (Otherwise, {{math|supp(''δ''<sub>''x''</sub>)}} is the closure of {{math|{''x''}<nowiki/>}} in {{math|(''X'', ''T'')}}.) Furthermore, {{math|''δ''<sub>''x''</sub>}} is the only probability measure whose support is {{math|{''x''}<nowiki/>}}.
* If {{math|''X''}} is {{math|''n''}}-dimensional [[Euclidean space]] {{math|'''R'''<sup>''n''</sup>}} with its usual {{math|''σ''}}-algebra and {{math|''n''}}-dimensional [[Lebesgue measure]] {{math|''λ''<sup>''n''</sup>}}, then {{math|''δ''<sub>''x''</sub>}} is a [[singular measure]] with respect to {{math|''λ''<sup>''n''</sup>}}: simply decompose {{math|'''R'''<sup>''n''</sup>}} as {{math|1=''A'' = '''R'''<sup>''n''</sup> \ {''x''}<nowiki/>}} and {{math|1=''B'' = {''x''}<nowiki/>}} and observe that {{math|1=''δ''<sub>''x''</sub>(''A'') {{=}} ''λ''<sup>''n''</sup>(''B'') = 0}}.
* The Dirac measure is a [[σ-finite measure|sigma-finite measure]].

==Generalizations==
A [[discrete measure]] is similar to the Dirac measure, except that it is concentrated at countably many points instead of a single point. More formally, 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]].

==See also==
* [[Discrete measure]]
* [[Dirac delta function]]

==References==
*{{cite book |title=Treatise on analysis, Part 2 |chapter=Examples of measures |page=100 |first=Jean |last=Dieudonné |chapter-url=https://books.google.com/books?id=7rCKMyHfd_EC&pg=PA100 |isbn=0-12-215502-5 |year=1976 |publisher=Academic Press}}
*{{cite book |title=Harmonic analysis and applications|first=John |last=Benedetto |chapter=§2.1.3 Definition, {{math|''δ''}} |chapter-url=https://books.google.com/books?id=_SCeYgvPgoYC&pg=PA72 |page=72 |isbn=0-8493-7879-6 |year=1997 |publisher=CRC Press}}

{{Measure theory}}

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[[Category:Measures (measure theory)]]