Integral of Positive Measurable Function as Limit of Integrals of Positive Simple Functions

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

Let $f: X \to \overline \R \in \MM_{\overline \R}^+$ be a positive $\Sigma$-measurable function.

Let $\sequence {f_n}_{n \mathop \in \N} \in \EE^+$, $f_n: X \to \R$ be a sequence of positive simple functions such that:


 * $\ds \lim_{n \mathop \to \infty} f_n = f$

where $\lim$ denotes a pointwise limit.

Then:


 * $\ds \int f \rd \mu = \lim_{n \mathop \to \infty} \int f_n \rd \mu$

where the integral signs denote $\mu$-integration.