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

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Theorem

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

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

Let $\left({f_n}\right)_{n \in \N} \in \mathcal{E}^+$, $f_n: X \to \R$ be a sequence of positive simple functions such that:

$\displaystyle \lim_{n \to \infty} f_n = f$

where $\lim$ denotes a pointwise limit.


Then:

$\displaystyle \int f \, \mathrm d\mu = \lim_{n \to \infty} \int f_n \, \mathrm d\mu$

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


Proof


Sources