Integral of Positive Measurable Function Extends Integral of Positive Simple Function

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Theorem

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

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


Then $\displaystyle \int f \, \mathrm d\mu = I_\mu \left({f}\right)$, where:

$\displaystyle \int \cdot \, \mathrm d\mu$ denotes the $\mu$-integral of positive measurable functions
$I_\mu$ denotes the $\mu$-integral of positive simple functions


That is, $\displaystyle \int \cdot \, \mathrm d\mu \restriction_{\mathcal{E}^+} = I_\mu$, using the notion of restriction, $\restriction$.


Proof


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