Measurable Function is Pointwise Limit of Simple Functions

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

Let $f: X \to \overline{\R}$ be a $\Sigma$-measurable function.

Then there exists a sequence $\left({f_n}\right)_{n \in \N} \in \mathcal E \left({\Sigma}\right)$ of simple functions, such that:


 * $\forall x \in X: f \left({x}\right) = \displaystyle \lim_{n \to \infty} f_n \left({x}\right)$

That is, such that $f = \displaystyle \lim_{n \to \infty} f_n$ pointwise.

The sequence $\left({f_n}\right)_{n \in \N}$ may furthermore be taken to satisfy:


 * $\forall n \in \N: \left\vert{f_n}\right\vert \le \left\vert{f}\right\vert$

where $\left\vert{f}\right\vert$ denotes the absolute value of $f$.