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 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.