# Definition:Dirac Measure

## Definition

Let $\left({X, \Sigma}\right)$ be a measurable space.

Let $x \in X$ be any point in $X$.

Then the **Dirac measure at $x$**, denoted $\delta_x$ is the measure defined by:

- $\delta_x: \Sigma \to \overline \R, \ \delta_x \left({E}\right) := \begin{cases}0 & \text{if } x \notin E \\ 1 & \text{if } x \in E\end{cases}$

where $\overline \R$ denotes the extended set of real numbers.

That $\delta_x$ actually is a measure is shown on Dirac Measure is Measure.

In fact, Dirac measure is a probability measure.

## Also known as

Alternatively, the **Dirac measure at $x$** may be called **Dirac's delta measure at $x$** or **unit mass at $x$**.

In physics, this measure is often (very informally) treated as a special function.

This obfuscates the rigid mathematical foundations the **Dirac measure** lies in, and thence should *always be avoided*.

## Source of Name

This entry was named for Paul Adrien Maurice Dirac.

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $4.7 \ \text{(i)}$