 Uniqueness theorem for Laplace transforms of nonnegative random variables

The Laplace transform of a nonnegative random variable determines its probability law. Indeed, $U=e^{-X}$ takes values in $(0,1]$, and $\mathbb{E}[U^n]={\psi }_X(n)$. Thus the transform determines every moment of $U$; moment determinacy on a compact interval determines the law of $U$, and the injective inverse $X=-\log U$ determines the law of $X$.