= Dirac measure
{c}
{title2=$\delta_x$}
{wiki}
On a <measurable space> $(X,\mathcal F)$, the <Dirac measure> at $x\in X$ assigns unit mass according to whether a <measurable set> contains that point:
$$
\delta_x(A)=\mathbf1_A(x)=\begin{cases}1,&x\in A,\\0,&x\notin A,\end{cases}\qquad A\in\mathcal F.
$$
Here $\mathbf1_A$ is the <indicator function>. This is a <probability measure>, since $\delta_x(X)=1$. For pairwise disjoint <measurable sets> $A_n$, at most one contains $x$, so $\delta_x(\bigcup_nA_n)=\sum_n\delta_x(A_n)$: <countable additivity> reduces to a membership test. The definition works even when the <singleton> $\{x\}$ is not a <measurable set>. https://terrytao.wordpress.com/2009/01/01/245b-notes-0-a-quick-review-of-measure-and-integration-theory/[Terence Tao, 245B notes 0, Example 7: Dirac measures]{ref}
To assert that its <support of a measure> is exactly $\{x\}$, assume that $X$ is a <metric space> and $\mathcal F$ is its <Borel sigma-algebra>. The <singleton> $\{x\}$ is then a <closed set> and a <Borel set>, with $\delta_x(\{x\})=1$. Every open <neighbourhood> of $x$ has mass one; every other point $y$ has an open <ball> of radius less than its distance from $x$, with mass zero. Thus $\operatorname{supp}\delta_x=\{x\}$. An abstract <measurable space> has no specified <topology>, so this topological conclusion is not part of the bare definition. Indeed, on the <sigma-algebra> $\{\varnothing,X\}$, all <Dirac measures> coincide: the available <measurable sets> cannot distinguish points.
The <Lebesgue integral> against a <Dirac measure> is evaluation:
$$
\int_X f\,d\delta_x=f(x).
$$
This holds for every nonnegative <measurable function>, allowing an infinite value, and for real or complex <measurable functions> satisfying $|f(x)|<\infty$, which are precisely the <integrable functions> for $\delta_x$. For a nonnegative <simple function>, only the cell containing $x$ contributes to its <integral>; increasing <simple functions> and the <monotone convergence theorem> extend the identity. Splitting into positive and negative parts, and then real and imaginary parts, gives the remaining cases. This is why a <Dirac measure> turns <integration> into a point query.
For a <measurable function> $T:(X,\mathcal F)\to(Y,\mathcal G)$, its <pushforward measure> satisfies
$$
(T_*\delta_x)(B)=\delta_x(T^{-1}(B))=\mathbf1_B(T(x)),\qquad T_*\delta_x=\delta_{T(x)}.
$$
Consequently $\int_Y g\,d(T_*\delta_x)=g(T(x))$ for nonnegative or integrable <measurable functions> $g$. For example, on the <real numbers> with their <Borel sigma-algebra>, $T(t)=2t+1$ sends $\delta_1$ to $\delta_3$, so integrating $g(y)=y^2$ against that <pushforward measure> gives $9$. https://terrytao.wordpress.com/2009/01/01/245b-notes-0-a-quick-review-of-measure-and-integration-theory/[Terence Tao, 245B notes 0, Example 10 and Theorem 2(8): pushforward and change of variables]{ref}
Finite weighted sums of <Dirac measures> give useful <measures> with finitely many point masses. For finite nonnegative weights $w_j$,
$$
\mu=\sum_{j=1}^m w_j\delta_{x_j},\qquad\mu(A)=\sum_{j=1}^m w_j\mathbf1_A(x_j),\qquad\int f\,d\mu=\sum_{j=1}^m w_jf(x_j)
$$
for nonnegative <measurable functions> $f$, or whenever $\sum_jw_j|f(x_j)|<\infty$, with zero-weight terms omitted. Repeated locations have their weights added. On the <real numbers>, this is the finite version of the <measure> underlying the <left-continuous cumulative function of an atomic measure>; a positive-weight <singleton> is an <atom of a measure>. Unit weights at the distinct points of a <finite set> recover <counting measure> on that <set>. https://terrytao.wordpress.com/2010/09/25/245a-notes-3-integration-on-abstract-measure-spaces-and-the-convergence-theorems/[Terence Tao, 245A notes 3, Exercises 21 and 25: representation by weighted Dirac measures]{ref}
The same <function> can have sharply different <integrals> for different <measures>. On $[0,2]$ with its <Borel sigma-algebra>, define
$$
f(t)=\begin{cases}5,&t=1,\\t+1,&t\ne1.\end{cases}\qquad\int_{[0,2]}f\,d\delta_1=5,\qquad\int_{[0,2]}f\,d\lambda=4,\qquad\int_{[0,2]}f\,d\#=\infty.
$$
Here $\lambda$ is <Lebesgue measure> and $\#$ is <counting measure>. The <singleton> $\{1\}$ is a <Lebesgue null set>, so its changed value leaves the <Lebesgue integral> equal to $\int_0^2(t+1)\,dt=4$. For <counting measure> on the whole <interval>, $f\geq1$ on an <infinite set>, forcing an infinite <integral>. On the smaller <measure space> $D=\{0,1,2\}$ with its <power set> and <counting measure>, the restricted <function> instead has <integral> $1+5+3=9$.
\Image[/counting-dirac-measures-evaluation-and-integration.png]
{title=An isolated value contributes differently to <Dirac measure>, <counting measure>, and <Lebesgue measure>}
{description=Original analytical illustration of the specified <function>, with an open circle at the excluded value $2$ and a filled point at the actual value $f(1)=5$. The finite sum uses $D=\{0,1,2\}$; the other three <integrals> use the same ambient <interval> $[0,2]$. The drawn area is the <Lebesgue integral>, and no density curve is assigned to the <Dirac measure>.}
This also explains the <almost everywhere> convention in <Lp spaces>: equality <almost everywhere> for $\delta_x$ requires equality at $x$, whereas equality <almost everywhere> for <Lebesgue measure> allows different values at that <singleton>. A point value therefore cannot be read unambiguously from an equivalence class in an <Lp space> formed using <Lebesgue measure>.
On the <real numbers>, a <Dirac measure> gives the <Dirac delta distribution> by $\langle\delta_x,\varphi\rangle=\int\varphi\,d\delta_x=\varphi(x)$ for every <test function>. The name <Dirac delta function> does not mean an ordinary <function> that is zero away from $x$ and infinitely high at $x$. Such a nonnegative <measurable function> still has <Lebesgue integral> zero, because its nonzero values occur on a <Lebesgue null set>. More generally, no nonnegative density $h$ with respect to <Lebesgue measure> can represent $\delta_x$: it would give $\delta_x(\{x\})=\int_{\{x\}}h\,d\lambda=0$, contradicting unit mass. The <measure> and its associated <distribution> act on different kinds of inputs, namely <measurable sets> and <test functions>, but express the same point evaluation. https://terrytao.wordpress.com/2009/04/19/245c-notes-3-distributions/[Terence Tao, 245C notes 3: Radon measures, test functions, and the Dirac delta distribution]{ref}
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