This question was motivated by another question in this site.
As explained in that problem (and its answers), if $\displaystyle f$ is continuous on $\displaystyle [0,1]$ and $\displaystyle \int_0^1 f(x)p(x)dx=0$ for all polynomials $\displaystyle p$, then $\displaystyle f$ is zero everywhere.
Suppose we remove the restriction that $\displaystyle f$ is continuous.
Can we conclude from $\displaystyle f\in L^1([0,1])$ that $\displaystyle f$ is zero almost everywhere?
(This should be terribly standard. My apologies, I am rusty of late.)