Integral of Positive Simple Function is Additive

Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f, g: X \to \R$, $f, g \in \EE^+$ be positive simple functions.


Then $\map {I_\mu} {f + g} = \map {I_\mu} f + \map {I_\mu} g$, where:

$f + g$ is the pointwise sum of $f$ and $g$
$I_\mu$ denotes $\mu$-integration


This can be summarized by saying that $I_\mu$ is additive.


Proof


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Sources

  • 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $9.3 \ \text{(iii)}$
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