Dirac Measure is Probability Measure

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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.

$\blacksquare$