If $f \in L^1[0,1]$ satisfies $\int_{0}^1f(x)e^{-2\pi i n x}dx = 0$ for $n \in \mathbb{N}$ ; is $f$ zero a.e?
Here $L^1[0,1]$ means the set of measurable functions $g : \mathbb{R} \rightarrow \mathbb{C}$ such that $\int_{0}^1|g(x)|dx < \infty$.
This result is true if $f \in L^2[0,1]$ as $\{e^{2\pi i nx} \}_{n=-\infty}^{n = \infty}$ is an orthonormal basis for $L^2[0,1]$.