Dirac measure

Definition

Let $X$ be a set, let $\mathcal{F}$ be a σ-algebra on $X$, and let $x \in X$. The Dirac measure at $x$ is the function $\delta_x \colon \mathcal{F} \to \{0, 1\}$ with $$\delta_x(A) = \begin{cases} 1 & \text{if } x \in A, \\ 0 & \text{if } x \notin A. \end{cases}$$

Remarks

$\delta_x$ is a probability measure. In terms of the indicator function, $\delta_x(A) = \mathbf{1}_A(x)$. The integral of a measurable function $f$ with respect to $\delta_x$ is $f(x)$. Each probability measure on a countable set $X$ with the power set as σ-algebra is a sum $\sum_{x \in X} p(x) \, \delta_x$ with weights $p(x) \ge 0$ of sum $1$.

Explanations

L1 human

The Dirac measure puts all the probability on one point. It describes an experiment whose result is certain, such as a coin with heads on both sides.