I am looking for an answer to the question: Why $\hat{\mu}\in L^2(\mathbb{R}^n)$ implies $\mu$ absolutely continuous?.
Namely, let $\mu$ be a compactly supported probability measure (or a finite measure for that matter) in $\mathbb{R}^n$ and suppose that $\widehat{\mu}\in L^2\left(\mathbb{R}^n\right)$. Show that $\mu$ is then absolutely continuous with respect to the Lebesgue measure.
The only answer to the aforementioned post (see here) claims that the conclusion is trivial as long as we know that the measure $\mu$ has "$L^2$ Lebesgue density". I have not managed to find material which would discuss the connection between the $L^p$ Lebesgue density and absolute continuity of a measure w.r.t. the Lebesgue measure. Therefore I am looking for a proof or a reference of the original claim and reference(s) to better understand the connection between the Lebesgue density and absolute continuity. Thanks!