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Transactions of the American Mathematical Society

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Fatou properties of monotone seminorms on Riesz spaces


Author: Theresa K. Y. Chow Dodds
Journal: Trans. Amer. Math. Soc. 202 (1975), 325-337
MSC: Primary 46A40
DOI: https://doi.org/10.1090/S0002-9947-1975-0355520-5
MathSciNet review: 0355520
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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 \vert f\vert\} $]> for <![CDATA[$ 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.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0355520-5
Keywords: Riesz spaces, monotone seminorms, Fatou properties, strong Egoroff property, order topology, relative uniform topology
Article copyright: © Copyright 1975 American Mathematical Society

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