Integral with respect to Discrete Measure

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

Let $\displaystyle \mu = \sum_{n \in \N} \lambda_n \delta_{x_n}$ be a discrete measure on $\left({X, \Sigma}\right)$.

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

Then:


 * $\displaystyle \int f \, \mathrm d\mu = \sum_{n \in \N} \lambda_n f \left({x_n}\right)$

where the integral sign denotes $\mu$-integration.