Integral of Positive Simple Function is Additive

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Theorem

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

Let $f,g: X \to \R$, $f,g \in \mathcal{E}^+$ be positive simple functions.


Then $I_\mu \left({f + g}\right) = I_\mu \left({f}\right) + I_\mu \left({g}\right)$, 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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