Let $\mu$ be a positive ($\sigma$-finite) measure on $\mathbb{R}$ absolutely continuous w.r.t. Lebesgue measure. I am looking for a function $0\neq f\in L^p(\mathbb{R},\mu)$, for a fixed $p\in[1,\infty)$, such that its Fourier transform $\hat f$ is supported on a set of finite measure, $$ |\mathbb{R}\setminus\hat f^{-1}(0)|<\infty. $$ There are no further restrictions on $\mu$.
Question: Does such a function $f$ exist?
Thank you.
Discussion: Since there are no restrictions on the growth of $\mu$ at infinity, I cannot hope that $\hat f$ is compactly supported, since Paley-Wiener functions cannot decay too fast. But finite measure support is a much weaker condition, I suppose.
One typical construction of functions supported on a set of finite measure is taking an infinite sum of shrunk and shifted copies of a single function of compact support. But since the union of supports of these copies has finite measure, such a system is not dense in $L^2$, and it is unclear how to make up a function with desired properties out of such building blocks.
If $\mu$ is too "weak" then a function $f\in L^p(\mathbb{R},\mu)$ may be very bad, so one may wonder what the Fourier transform of such an $f$ is. I am interested in such $f$ for which Fourier transform makes sense, for instance, $f\in L^2(\mathbb{R})$.