= L2 density from a square-integrable Fourier transform
{c}
{title2=$\widehat\mu\in L^2\ \Longrightarrow\ d\mu=g\,dx,\ g\in L^2$}
If a <finite measure> on $\mathbb R^n$ has square-integrable <Fourier transform>, the <Plancherel theorem> gives an $L^2$ inverse transform $g$. Pairing with every <Schwartz function> and using <Fourier inversion> identifies the measure with $g\,dx$. This establishes absolute continuity, rather than presuming it. For a positive measure, $g\ge0$ and $g\in L^1$ as well. Multiplication by $f\in L^\infty(\mu)$ yields $\|\widehat{f\,d\mu}\|_2\le\|\widehat\mu\|_2\|f\|_{L^\infty(\mu)}$.
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