Von Neumann and
Zaanen have studied measure theoretic properties of collections of sets which
satisfy weaker axioms than those of a ring. In this paper it is shown that the
von Neumann axioms for a half ring of sets and the Zaanen axioms for a
semi-ring of sets can be weakened without loss of their measure theoretic
significance.
An investigation of the geometrical structure of a collection ℛ of convex sets
which satisfy either von Neumann’s, Zaanen’s or our weaker axioms is conducted.
Principally we extend some earlier results by showing that under rather mild
restrictions, sets of such collections are polyhedral. After imposing the additional
condition that ℛ∖{ϕ} be a neighborhood base for a linear topology, we
prove that if ℛ is a semi-ring in the earlier sense then the topology induced
by ℛ is a so called weak topology and conversely every weak topology has
such a neighborhood base. Finally we characterize subspaces of the Banach
space (c0) as the only Banach spaces which have a neighborhood base of
convex sets which together with the null set form a half ring (in the weaker
sense).
