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- TL;DR Summary
- I am studying a theorem in probability that the Laplace transform of a (nonnegative) random variable determines the law of that random variable, which is equivalent to its injectivity. The book only treats Laplace transforms of nonnegative random variables, but since the moment generating function (mgf) is in fact the two sided Laplace transform, and since mgfs exist for arbitrary sign random variables, I wonder how to extend the result to say the two-sided Laplace transform is also injective.

I will omit the theorem and its proof here, since it would mean a lot of typing. But the relevant part of the proof of the theorem is that we are considering the set ##H## of functions consisting of ##\psi_\lambda(x)=e^{-\lambda x}## for ##x\geq 0## and ##\lambda\geq 0##. We extend the functions by continuity so they are defined on all of ##[0,\infty]##, namely we put ##\psi_\lambda(\infty)=0## if ##\lambda>0## and ##\psi_0(\infty)=1##. Having a compact set ##[0,\infty]##, we can apply the Stone-Weierstrass theorem to say that ##H## is dense in ##C([0,\infty])##.

I have seen another proof of the injectivity of the Laplace transform (not related to probability), but that also uses the Stone-Weierstrass theorem. So how would one go about showing the two-sided Laplace transform is injective? I guess one would like to consider ##[-\infty,\infty]## and apply the Stone-Weierstrass theorem also, but I'm unsure how one would modify ##H##.

I have seen another proof of the injectivity of the Laplace transform (not related to probability), but that also uses the Stone-Weierstrass theorem. So how would one go about showing the two-sided Laplace transform is injective? I guess one would like to consider ##[-\infty,\infty]## and apply the Stone-Weierstrass theorem also, but I'm unsure how one would modify ##H##.