here is a qualifying exam problem and my best attempt thus far: (This is a previous problem, I'm not taking it right now.)
Problem: if $(\mathbb{R},\mu)$ is a finite Borel measure and $\phi(t)=\int e^{ixt}\,d\mu(x)$ is its Fourier transform if furthermore $\mu(\{x\})=0$ for any $x\in \mathbb{R}$, prove that $$\frac{1}{T}\int_{-T}^T|\phi(t)|^2\,dt\rightarrow 0$$ as $T\rightarrow\infty$.
My attempt is to write $$\int_{-T}^T|\phi(t)|^2\,dt=\int\int_{-T}^Te^{-itx}\phi(t)\,dt\,d\mu(x).$$
While the inversion theorem says that $$\int_a^b\int_{-T}^Te^{-itx}\phi(t)\,dt\,dx\rightarrow 2\pi\mu([a,b])$$ as $T\rightarrow\infty$ and somehow go from here. But I'm kind of stuck. Please help me!! Thanks!!
Oh, actually this follows from An identity for Fourier transform of measure