Integral of Integrable Function is Additive

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Theorem

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

Let $f, g: X \to \overline \R$ be $\mu$-integrable functions.

Suppose that their pointwise sum $f + g$ is well-defined.


Then:

$\displaystyle \int f + g \rd \mu = \int f \rd \mu + \int g \rd \mu$


Proof


Sources