Exponential characterization by a uniform random split
= Exponential characterization by a uniform random split
Write $h(y)=\int_y^\infty g(t)/t\,dt$. The two pieces in a uniform split are independent exactly when
$$
-h'(y+z)=h(y)h(z).
$$
Setting $z=0$ shows that $h(y)=ce^{-cy}$ 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 $c^2xe^{-cx}$.