This is one of the problem in our past comprehensive exams. I don't mind getting full solution.
Suppose $f$ is a bounded, measurable function on $[0,1]$, $\epsilon>0,$ and for all $x>\epsilon\,$ one has
$$0=\int_0^1 f(s )\exp(-xs)ds$$
Show that $f=0$ almost everywhere.
Someone gave me a hint to solve the problem using Urysohn's lemma. I am not totally comfortable with that lemma. I have a hunch that we can prove this along the line of Fourier analysis. I am not that sure on this approach either. I don't even know how to get started.