Assume the function $u\to \mathbb E[e^{iuX}]$ is analytic in a nbhd of $0$ where $X$: $\Omega\to \mathbb R$ is a random variable.
Now I want to conclude that the space of polynomial, denoted by $\mathbb T$, is dense in $L^2(\mathbb R,\,\mathbb P\circ X^{-1})$ with $L^2$ norm, where $\mathbb P$: $\Omega\to [0,1]$ is a probability measure.
I think as long as $\mathbb P\circ X^{-1}$ forms a Radon measure then I am done. But I am not sure about this since $X$ maybe pretty bad. I think I should use the property that $u\to \mathbb E[e^{iuX}]$ is analytic but I can't see how...
Any help is really welcome!