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.


Proof


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Sources

  • 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 9$: Problem $7$
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