Boundedness of Nonadditive Quantum Measures

We deal with not necessarily additive functions acting on complete orthomodular posets and taking values in Hausdorff uniform spaces, where no algebraic structure is required. As a consequence, neither pseudo-additivity, nor monotonicity are meaningful notions in this setting. Conditions ensuring their boundedness are exhibited in terms of some mild continuity properties. Such conditions are satisfied, in particular, by completely additive measures on projection lattices of von Neumann algebras. Hence, among other things, our main result provides a version in the generalized nonadditive quantum setting of the so-called boundedness principle in classical and quantum measure theory.