 Moment-generating function of an independent product

If $U$ and $X$ are independent real random variables, then the moment-generating function of their product is $M_{UX}(t)={\mathbb{E}}_U[M_X(tU)]$ wherever the expression is finite. This follows by conditioning on $U$: the conditional law of $X$ is unchanged by independence, so $\mathbb{E}(e^{tUX}\mid U)=M_X(tU)$. For $U$ uniform on $(0,1)$, the formula becomes $M_{UX}(t)={\int }_0^1{M}_X(tu)du$. If $M_X$ is finite on a neighborhood of zero, this product also has a finite moment-generating function on a neighborhood of zero, because the bounded factor $U$ keeps $tu$ between zero and $t$.