Vol. 17, No. 1, 1966

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Half rings in linear spaces

P. H. Maserick

Vol. 17 (1966), No. 1, 137–148

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 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 is a semi-ring in the earlier sense then the topology induced by 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).

Mathematical Subject Classification
Primary: 28.10
Received: 25 July 1961
Published: 1 April 1966
P. H. Maserick