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.
$\blacksquare$