Is Plancherel's theorem on a Weighted $L^2$-space valid?

Question

In Rudin, Real and complex, (Plancherel's Theorem) $$\left\|f\right\|_{L^2(dx)}=\left\|\widehat{f}\right\|_{L^2(dx)},\quad f\in L^2(\mathbb{R})$$ with $dx$ Lebesgue measure.

In Function, spaces, and expansion by Christiansen,

Question 1. Is Plancherel's theorem on a Weighted $L^2$-space valid? i.e. $\left\|f\right\|_{L^2(d\mu)}=\left\|\widehat{f}\right\|_{L^2(d\mu)}$ with $d\mu=r(x)dx$? (Here, $f\in L_r^2(\mathbb{R})=L^2(d\mu)$)

Answer 1

Of course not. Try $r(x)=e^{-x^2/2}$ and $f(x)=(b^2+x^2)^{-1}$ as a counterexample.