During one of the problems in Rudin I was asked to show $f=0$ a.e. Here $f$ satisfies this condition:
$f(x)=\frac{1}{x}\int^{x}_{0}f(t)dt$ almost everywhere and is in $L^{p}(0,\infty)$. So constant functions would not work. I tried to prove by contradiction, and a few imaginary counter-examples' failure convinced me this is true. But what is a good way of proving this statement? Since we know $f\in L^{p}$ I am thinking about using Holder's inequality, but in our case it is difficult to apply (since the other side is larger ). We can assume $f\in C_{c}(0,\infty)$ since this is dense in $L^{p}$, but I still do not know how to prove this statement.