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> [!definition] Definition. ([[Dirac measure]])
> Let $(X, \Sigma)$ be a [[σ-algebra|measurable space]]. Let $x \in X$. The **Dirac measure** corresponding to $x$ $A$ [[characteristic function|is]] $\delta_{x}(E)=\chi_{E}(x)=\begin{cases}
1 & x \in E \\
0 & x \not \in E.
\end{cases}$
^definition
> [!justification]
> $\delta_{x}$ is clearly a measure: certainly $\delta_{x}(\emptyset)=0$, and $\begin{align}
\delta_{x}(\bigsqcup_{k=1}^{\infty} E_{k})=\begin{cases}
1 & x \text{ is in one of the }E_{k} \\
0 & \text{else}
\end{cases}
\end{align}$
while $\sum_{k=1}^{\infty}\delta_{x}(E_{k})=\begin{cases}
1 & x \text{ is in one of the }E_{k} \\
0 & \text{else}
\end{cases}$
where we have used that the $E_{k}$ are disjoint, so $x$ can be in at most one.
^justification
> [!basicproperties]
> - $\delta_{x}$ is a [[probability|probability measure]]
> - If $(\Omega, \mathcal{F}, \mathbb{P})$ is a [[probability|probability space]] and $X:\Omega \to E$ is a [[random variable]] with $\mathbb{P}(X=x)=1$ for some $x$, [[probability distribution|then]] $\mathbb{P}_{X}=\delta_{x}$.[^1]
> - (Sifting) $\int f \, d \delta_{x}=f(x)$. (exercise)
^properties
[^1]: Indeed, given $B$ one has $\mathbb{P}_{X}(A)=\mathbb{P}(X \in A)$$\mathbb{P}_{X}(B)=\mathbb{P}(X \in B)=\begin{cases}
1 & x \in B \\
0 & x \not \in B
\end{cases}=\delta_{x}(B).$Sorry for the notation clash regarding $X$ and regarding $E$.
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####
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#### References
> [!backlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM [[]]
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```