Maharam-Types and Lyapunov's Theorem for Vector Measures on Locally Convex Spaces without Control Measures

We formulate the saturation property for vector measures in lcHs as a nonseparability condition on the derived Boolean $\sigma$-algebras by drawing on the topological structure of vector measure algebras. We exploit a Pettis-like notion of vector integration in lcHs, the Bourbaki--Kluv\'anek--Lewis integral, to derive an exact version of the Lyapunov convexity theorem in lcHs without the BDS property. We apply our Lyapunov convexity theorem to the bang-bang principle in Lyapunov control systems in lcHs to provide a further characterization of the saturation property.