Integral with respect to Series of Measures

Theorem

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

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

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

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

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