Integral with respect to Dirac Measure

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

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

Let $f \in \mathcal{M}_{\overline{\R}}, f: X \to \overline{\R}$ be a measurable function.

Then:


 * $\displaystyle \int f \, \mathrm d\delta_x = f \left({x}\right)$

where the integral sign denotes the $\delta_x$-integral.