 Solution

SetThese are bounded random variables with values in $(0,1]$, and for every positive integer $n$,Part c shows that $U$ and $V$ have the same law. Since $x\mapsto e^{-x}$ is injective on $[0,\infty )$ with inverse $u\mapsto -\log u$, taking the pushforward measure by this inverse shows that $X$ and $Y$ have the same law. This proves the uniqueness theorem for Laplace transforms of nonnegative random variables.