Fatou properties of monotone seminorms on Riesz spaces

Abstract:

A monotone seminorm $\rho$ on a Riesz space $L$ is called $\sigma$-Fatou if $\rho ({u_n}) \uparrow \rho (u)$ holds for every $u \in {L^ + }$ and sequence $\{ {u_n}\}$ in $L$ satisfying $0 \leq {u_n} \uparrow u$. A monotone seminorm $\rho$ on $L$ is called strong Fatou if $\rho ({u_v}) \uparrow \rho (u)$ holds for every $u \in {L^ + }$ and directed system $\{ {u_v}\}$ in $L$ satisfying $0 \leq {u_v} \uparrow u$. In this paper we determine those Riesz spaces $L$ which have the property that, for any monotone seminorm $\rho$ on $L$, the largest strong Fatou seminorm ${\rho _m}$ majorized by $\rho$ is of the form: ${\rho _m}(f) = \inf \{ {\sup _v}\rho ({u_v}):0 \leq {u_v} \uparrow |f|\}$]> for $f \in L$. We discuss, in a Riesz space $L$, the condition that a monotone seminorm $\rho$ as well as its Lorentz seminorm ${\rho _L}$ is $\sigma$-Fatou in terms of the order and relative uniform topologies on $L$. A parallel discussion is also given for outer measures on Boolean algebras.