Let $E$ be a Lebesgue measurable set in $\mathbb{R}$ having measure $1$. Prove that there is an integer $n\neq0$ such that $\int_Ee^{inx}dx\neq0$.
This is an exercise problem. A hint is given: periodize with period $2\pi$. However, I still do not have an idea how to prove it. Any help will be appreciated.