Integral of Positive Measurable Function is Additive

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Theorem

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

Let $f, g: X \to \overline \R$, $f, g \in \MM_{\overline \R}^+$ be positive measurable functions.


Then:

$\ds \int \paren {f + g} \rd \mu = \int f \rd \mu + \int g \rd \mu$

where:

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


This can be summarized by saying that $\ds \int \cdot \rd \mu$ is (conventionally) additive.


Proof


Sources