Integral of Positive Measurable Function Extends Integral of Positive Simple Function

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

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

Then:
 * $\ds \int f \rd \mu = \map {I_\mu} f$

where:


 * $\ds \int \cdot \rd \mu$ denotes the $\mu$-integral of positive measurable functions
 * $I_\mu$ denotes the $\mu$-integral of positive simple functions

That is:
 * $\ds \int \cdot \rd \mu \restriction_{\EE^+} = I_\mu$

using the notion of restriction, $\restriction$.