fa.functional analysis – Integrable functions that does not satisfy the inversion formula

Let $fin L^1(mathbb{R})$ and suppose that ${zetain mathbb{R} : e^{2pi izeta x} f(x) notin L^1(mathbb{R})}$ is a null set (its Lebesgue measure is zero). Can we conclude that $f(x)=int_{mathbb{R}} hat{f}(zeta)e^{-2pi izeta x}dzeta$ almost every where?

Does there exist any Lebesgue integrable function $f$ for which ${zetain mathbb{R} : e^{2pi izeta x} f(x) notin L^1(mathbb{R})}$ enjoys non-zero Lebegsue measure?