I recently came across this problem, namely we are given a continuous function $f:\mathbb R\to\mathbb R$ such that
$\int_0^1f(u(x))\mathrm dx=0,\;\forall u\in C^0([0,1]):\int_0^1u(x)\mathrm d x=0.$
I am asked to prove that the same property holds for any $u\in L^\infty([0,1])$ such that $\int_0^1 u(x)\mathrm dx=0.$
My attempt is to exploit Lusin Theorem because my first thought is that an essentially bounded measurable function is nearly a continuous one and then use the property given for continuous function. However I' still having problems in figuring out the reasoning. Is my path correct or not? and how should I approach the problem? Thank you.
Edit This is a further thought that came to my mind. Is such an $f$ necessarily a linear map? Because of course linear maps do the jobs, but what about the converse? The thought came by noticing that if $u$ is a continuous function satisfying the hypothesis given, then so does $\lambda u$, and if $v$ is another such function, then $u+v$ is fine as well.
I didn't want to ask another question because it descends from the original problem i proposed. Hope it is ok to ask here. Bye.