The class $I_0$ for random increasing upper semicontinuous functions

Let $C$ be the convex cone $USC_*([0,1],\mathbf{R}_+)$ of increasing upper semicontinuous functions $g\colon[0,1]\to\mathbf{R}_+$. It is shown that the class $I_0(C)$ of distributions on $C$ without indecomposable factor is strictly included in the class of infinitely divisible distributions on $C$.