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

Source abstract

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