  Lyapunov's theorem is a classical result in convex analysis, concerning the
convexity of the range of nonatomic measures. Given a family of integrable
vector functions on a compact set, this theorem allows to prove the equivalence
between the range of integral values obtained considering all possible set
decompositions and all possible convex combinations of the elements of the
family. Lyapunov type results have several applications in optimal control
theory: they are used to prove bang-bang properties and existence results
without convexity assumptions. Here, we use the dual approach to the Baire
category method in order to provide a "quantitative" version of such kind of
results applied to a countable family of integrable functions.
