The condition on \(\sigma\)-convexity is met by the subsets of \(\sigma\)- additive and completely additive probability measures on an orthomodular poset. A convex subset of probability measures which is compact in a linear Hausdorff topology is \(\sigma\)-convex. Also it is shown that the collection of Jauch-Piron probability measures forms a \(\sigma\)-convex set provided that the orthomodular poset is a \(\sigma\)-complete lattice.
The main theorem states that a \(\sigma\)-convex subset of probability measures on an orthomodular poset yields a base norm on the linear span of this subset which is complete. This theorem is obtained as a consequence of a general result on completeness, also established here, which holds in the context of Hausdorff topological vector spaces.