Question
Let X be a random variable with characteristic function $\varphi$. Show that
$$ \mathbb{P}(X = a ) = \lim_{T \rightarrow \infty} \frac{1}{2T} \int_{-T}^T e^{-ita}\varphi(t)dt. $$
Attempt
Let $F(x) = \mathbb{P}(X<x)$; and $\varphi(t) = \mathbb{E}[e^{itX}]$
By Levy's inversion formula: $$ \mathbb{P}(X=a) = \lim_{\epsilon \downarrow 0} F(a+\epsilon)-F(a)= $$
$$ =\lim_{\epsilon \downarrow 0}\frac{1}{2 \pi} \lim_{T \rightarrow \infty} \int_{-T}^{T}\frac{e^{-i(a+\epsilon)t}-e^{-iat}}{it}\varphi(t)dt= $$
$$ =\frac{1}{2 \pi}\lim_{\epsilon \downarrow 0} \lim_{T \rightarrow \infty} \epsilon \int_{-T}^{T}e^{-iat}\frac{e^{-\epsilon t}-1}{\epsilon it}\varphi(t)dt $$
But then I don't know what to do. Can you give me any suggestions please? Thank you