Uniqueness theorem for Laplace transforms of nonnegative random variables
= 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 $X$.