Integral with respect to Series of Measures

Theorem
Let $\struct {X, \Sigma}$ be a measurable space.

Let $\ds \mu := \sum_{n \mathop \in \N} \lambda_n \mu_n$ be a series of measures on $\struct {X, \Sigma}$.

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


 * $\ds \int f \rd \mu = \sum_{n \mathop \in \N} \lambda_n \int f \rd \mu_n$

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