I think that it is well-known that a real valued function $f\in L_\infty[a,b]$ can be approximated by continuous functions $f_n$ with respect to the $L_1$-norm, i.e. $||f_n-f||_{L^1}\to0$, where the $f_n$ can be chosen so that $||f_n||_{L_\infty}\leq||f||_{L_\infty}$. EDIT: Here, I'm only using the ordinary Lebesgue-measure for functions $f:[a,b]\to\mathbb R$.
I know a proof using Lusin's and Tietze's Extension Theorem, but I don't want to reproduce the proof, because the proof is not necessary for my needs and quite technical. Therefore, I was looking for references for this result, but couldn't find any. Any help is very appreciated. Thank you in advance!