Let $g:R\to [0,\infty)$ is measurable such that $\int_R g(t)e^{nt} dt \leq M$ for all $n=1,2,... $ where $M$ is finite. To prove that $g=0$ a.e. I have no idea where to start with this problem, any hints will be appreciated.

1$\begingroup$ This seems to be (nice) homework. $\endgroup$ – Dieter Kadelka 2 days ago

$\begingroup$ Obviously false as stated. Are there missing quantifiers? $\endgroup$ – Steven Landsburg yesterday