Let $\mu$ be a positive finite measure on $\mathbb R$. Is it true that $\int_{\mathbb R} T \text{sinc}^2(Tx) d\mu(x) \sim\frac{\mu([-1/T,1/T])}{1/T}, \text{ as } T\to\infty?$ Here $\text{sinc}(x)=\frac{\sin x}{x}$ and the notation $f(T)\sim g(T)$ as $T\to\infty$ means that $\lim_{T\to\infty}f(T)/g(T)=c<\infty$.
Of course, this holds if $\mu$ is absolutely continuous (a classical theorem on summability kernels), where the limit of both sides exists and is finite. On the other hand, if $\mu$ is not absolutely continuous near $0$, then the RHS tends to infinity (by theory of differntiation of measures). How to prove that the LHS tends to infinity with the same asymptotics?