Dirac Measure is Probability Measure

Theorem

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

Let $x \in X$, and let $\delta_x$ be the Dirac measure at $x$.

Then $\delta_x$ is a probability measure.

Proof

By Dirac Measure is Measure, $\delta_x$ is a measure.

Also, $\delta_x \left({X}\right) = 1$ because $x \in X$.

Hence $\delta_x$ is a probability measure.

$\blacksquare$