In the following result: $$\frac{1}{2 \pi} \int_{-\infty}^{\infty} \Phi(u) \left[\int_a^{\infty} e^{-i u x}dx\right] du = \frac{1}{2} + \frac{1}{ \pi} \int_0^{\infty} \Re\left[\frac{e^{-i u a} \Phi(u)}{i u}\right] du$$ $\Phi(u)$ is a characteristic function and $a$ is a constant. I think $$\int_a^{\infty} e^{-i u x}dx$$ is divergent, because $\lim_{t \to \infty} e^{-\imath t} $ does not exist.
Question is, how does above result exist then? Even if I suppose that $\lim_{t \to \infty} e^{-\imath t} =0 $, I cannot get to the RHS.