Let $X$ denote a real-valued random variable with characteristic function $\phi$. Suppose that $g$ is a real-valued function on $\mathbb{R}$ that has the representation
$\hspace{25mm}g(x) = \int_{-\infty}^{\infty}G(t)\exp(itx)\,dt$
for some $G$ satisfying
$\hspace{25mm}\int_{-\infty}^{\infty}|G(t)|dt < \infty$
Prove that
$\hspace{15mm}$$E[g(X)] = \int_{-\infty}^{\infty}G(t)\phi(t) dt$
Attempt:
$\hspace{15mm}$ $E[g(X)] = \int_{-\infty}^{\infty}G(t)\exp(itX)\cdot p(x) dx$
Since $\int_{-\infty}^{\infty}\exp(itX)\cdot p(x) dx$ = $E[\exp(itX)] = \phi(t)$, we then have
$\hspace{15mm}$$E[g(X)] = \int_{-\infty}^{\infty}G(t)\phi(t) dt$