A generalization of Lyapunov's convexity theorem with applications in optimal stopping

Lyapunov proved that the range of $n$ finite measures defined on the same $\sigma$-algebra is compact, and if each measure $\mu _{i}$ also is atomless, then the range is convex. Although both conclusions may fail for measures on different $\sigma$-algebras of the same set, they do hold if the $\sigma$-algebras are nested, which is exactly the setting of classical optimal stopping theory.