Proving that MGF determines PDF when the PDF is defined for whole real line

Question

If two PDFs have the same moment generating function that converges in an open set around 0, then the PDFs are same.

This is a well known fact, but I can't find its proof. If the PDFs are defined for only non-negative values, the MGF is essentially Laplace transform and the uniqueness problem is just that of Laplace transform. However, if the PDFs are defined on the whole real line, the MGF is not Laplace transform. I guess the condition that the MGF converges around zero may take care of something in this case. Could someone provide the proof or reference for it?

Answer 1

If the random variable $X$ has pdf $f(x)$, $x\in \mathbb{R}$, then the moment generating function $M_X(t)$, provided it exists, is the two-sided Laplace transform of $X$. If $\mathcal{L}\{f(x)\}(t)$ is the (one-sided) Laplace Transform of $f(x)$, then

$$ M_X(-t)=\mathcal{L}\{f(x)\}(t)+\mathcal{L}\{f(-x)\}(-t) $$

and $M_X(t)$ is unique iff both $\mathcal{L}\{f(x)\}(t)$ and $\mathcal{L}\{f(-x)\}(-t)$ are unique. But we already know that the Laplace transform of a function is unique, provided it exists.