Integral with respect to Series of Measures

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

Let $\mu := \sum_{n \in \N} \lambda_n \mu_n$ be a series of measures on $\left({X, \Sigma}\right)$.

Then for all positive measurable functions $f: X \to \overline{\R}, f \in \mathcal{M}_{\overline{\R}}^+$:


 * $\displaystyle \int f \, \mathrm d\mu = \sum_{n \in \N} \int f \, \mathrm d\mu_n$

where the integral signs denote integration with respect to a measure.