Fatou Properties of Monotone Seminorms on Riesz Spaces

Theresa K. Y. Chow Dodds
1975 Transactions of the American Mathematical Society  
A monotone seminorm p on a Riesz space L is called a Fatou if p(a")tp(«) holds for every u e L and sequence {u"} in L satisfying 0 < un\u. A monotone seminorm p on L is called strong Fatou if p(u )tp(u) holds for every u e L and directed system {«"} in L satisfying 0 < «"tu. In this paper we determine those Riesz spaces L which have the property that, for any monotone seminorm p on L, the largest strong Fatou seminorm p m majorized by p is of the form: Pm(/) = inf {sup"p(up): 0 <upt\f\} for
more » ... 0 <upt\f\} for /ei. We discuss, in a Riesz space L, the condition that a monotone seminorm p as well as its Lorentz seminorm p¿ is o-Fatou in terms of the order and relative uniform topologies on L. A parallel discussion is also given for outer measures on Boolean algebras.
doi:10.2307/1997313 fatcat:o437unejyndkjktg2fqmw2v5va