A class of special subordinators with nested ranges

We construct, on a single probability space, a class of special subordinators $S^{(\alpha)}$, indexed by all measurable functions $\alpha: [0,1]\to [0,1]$. Constant functions correspond to stable subordinators. If $\alpha\leq \beta$, then the range of $S^{(\alpha)}$ is contained in the range of $S^{(\beta)}$. Other examples of special subordinators are given in the lattice case.