Dirac Measure is Probability Measure
Jump to navigation
Jump to search
Theorem
Let $\struct {X, \AA}$ 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, $\map {\delta_x} X = 1$ because $x \in X$.
Hence $\delta_x$ is a probability measure.
$\blacksquare$