Uniform split of a random total

Let $X\ge 0$ have density $g$, let $U$ be independent and uniform on $[0,1]$, and setThen

Write $h(y)={\int }_y^{\infty }g(t)/tdt$. The two pieces in a uniform split are independent exactly whenSetting $z=0$ shows that $h(y)=c{e}^{-cy}$ for some $c>0$. Thus the pieces are independent exactly when they are independent exponential variables of the same rate; equivalently, the total has gamma density $c^{2}x{e}^{-cx}$.