We know that if $X$ is a real-valued random variable with characteristic function, then $\varphi\in C^k(\mathbb R)$ and $$\varphi^{(k)}(t)=\operatorname E\left[({\rm i}X)^ke^{{\rm i}tX}\right]\tag1$$ for all $k\in\mathbb N_0$ with $\operatorname E\left[|X|^k\right]<\infty$.
Now I somehow need to use this in the opposite direction: I have a finite measure $\nu$ on $\mathbb R$ and a real-valued random variable $X$ with characteristic function $$\varphi(t):=\exp\left(\int e^{{\rm i}tx}-1\:{\rm d}\nu({\rm d}x)\right)\;\;\;\text{for }t\in\mathbb R.$$
Clearly, $\varphi\in C^\infty(\mathbb R)$. What I need now, is showing that $$\operatorname E\left[X\right]=\int\nu({\rm d}x)x\tag2$$ and $$\operatorname{Var}\left[X\right]=\int\nu({\rm d}x)x^2\tag3.$$ While $(2)$ and $(3)$ immediately follow from $(1)$, I'm not sure whether I'm allowed to apply $(1)$, since I don't know whether $\operatorname E\left[|X|^k\right]<\infty$ for $k=1,2$ a priori.