On the Real Analysis - Modern Techniques and Their Application (second edition) by Gerald Folland, page 47 i found this theorem: "Let $f$ a measurable function. Then exists a sequence $(\phi_n)_{n \in \mathbb{N}}$ of simple functions such that: (a) $\phi_n \to f$ pointwise; (b) $\phi_n \to f$ uniformly on any set on which $f$ is bounded".
I need a proof step by step of this theorem. Can someone suggest me a more didactic book?