Definition:Dirac Measure

From ProofWiki
Jump to: navigation, search

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