Dirac Measures

Dirac Measure

Definition: Dirac Measure

Let $(X,\Sigma )$ be a measurable space and let $p\in X$.

The Dirac measure on $(X,\Sigma )$ for $p$ is the function ${\delta }_p:\Sigma \to \{0,1\}$ defined for each $S\in \Sigma$ as

$${\delta }_p(S)\stackrel{\text{def}}{=}\{\begin{array}{l}1\text{if }p\in S\\ 0\text{if }p\notin S\end{array}$$

Theorem: Dirac Measure is a Measure

The Dirac measure ${\delta }_p$ is a measure on $(X,\Sigma )$.

PROOF

We need to prove the following:

• (1) ${\delta }_p(\varnothing )=0$
• (2)

Proof of (1):

Since $p\notin \varnothing$, we have ${\delta }_p(\varnothing )$ as per definition of ${\delta }_p$.