Integral with respect to Series of Measures
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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 $7$