Let $X$ be some random variable with characteristic function $\hat{X}$. Show that $X$ has no point masses if and only if
$$ \lim_{T\to +\infty}\frac1{2T}\int_{-T}^T\left\lvert \hat{X}(t)\right\rvert^2\mathrm dt=0. $$
I think I have to consider the probability $P(X − Y = 0)$ where $Y$ is independent of $X$ and $X=Y$ in distribution, but I don't know how proceed.