I am trying to prove the following fact:
Let $X$ be real-valued random variable and $\phi$ its characteristic function. Then for every real $a$ the following holds: $P(X=a) = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} e^{-ita}\phi(t)dt$.
I've reduced this identity to the following one:
$P(X=a) = \lim_{T \to \infty} \int_{\mathbb{R}} \frac{\sin T(x-a)}{T(x-a)}dP_X(x)$.
Not it sure now why it should hold. I would be grateful for your suggestions or ideas.
Thanks