Fatou Properties of Monotone Seminorms on Riesz Spaces

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 /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. 1. Notation. It is convenient to first introduce some notation. Indices from a countable set will be denoted by m, n, k, •••, from an arbitrary set by k,v, p, •••. Let X be any partially ordered set. If xEX, and if a sequence {xn } in X satisfies xn\x, and if for each n a sequence {xnk: k = 1, 2, *••} satisfies xnk\kxn, then we will write jcnfctjcntx. If x E X, and if an upwards directed system {xv: v E \] } satisfies xv\x, and if for each vE\] an index set Kv and an upwards directed system [xK :kvEK„} satisfy xK tK eK x, then we will write x„ t¡c„tx 2. Outer measures on Boolean algebras. A real function 0 on a Boolean algebra B is called a finitely additive measure if (i) 0 < 0(a) < °° for all aEB, (ii) 0(aVo) =0(a) + 0(A) whenever a, b EB and a, b are disjoint, (iii) 0(1) =£ 0. A finitely additive measure 0 on B is called countably additive if 0(V7a«) = ^T 0(a«) f°r every mutually disjoint countable subset {ax, a2, •••} of 5 such that VTM0« exists. A finitely additive measure 0 on B is called purely finitely additive if every countably additive measure 0' such that 0' < 0 is identically zero. In [11], K. Yosida and E. Hewitt proved that every Received by the editors March 20, 1971 and, in revised form, April 9, 1973. AMS (MOS) subject classifications (1970). Primary 46A40.