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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

\[ \delta_p(S) \overset{\text{def}}{=} \begin{cases} 1 \qquad \text{if } p \in S \\ 0 \qquad \text{if } p \notin S\end{cases} \]

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 \(\delta_p\).