---
tag: 099SXAG014
kind: definition
name: Dirac measure
msc: [28A12]
defines: 0ACRGYNHSR
uses: [04WSW8T3SY, 0VGMEBVW1M, 0VSB0XF4G7]
sources:
  - url: https://en.wikipedia.org/wiki/Dirac_measure
    locator: Definition
---

## Definition

Let $X$ be a set, let $\mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]] 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|0VGMEBVW1M]]. In terms of the [[indicator function|0VSB0XF4G7]], $\delta_x(A) = \mathbf{1}_A(x)$.
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.