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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!

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  • $\begingroup$ You can use that the simple functions are dense in $L^1$. In case you are looking at some regular borel measure, you can approximate simple functions by finite linear combinations of indicator functions of open balls, which you can easily approximate by continuous functions (linear interpolation). $\endgroup$ – Severin Schraven Feb 10 '19 at 13:58
  • $\begingroup$ Thanks, but as I said I would like to just cite this result and not include a proof. Do you have a reference? $\endgroup$ – sranthrop Feb 10 '19 at 14:01
  • $\begingroup$ I cannot think of a reference on the top of my head and I'm not in my office to check. $\endgroup$ – Severin Schraven Feb 10 '19 at 14:05

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