Submeasures and locally solid topologies on Riesz spaces. (English) Zbl 0601.46006

Labuda, Iwo

Math. Z. 195, 179-196 (1987).

The main tool in this paper is a general representation theorem, presented in Section 2, which gives necessary and sufficient conditions in order that an Archimedean Riesz space L admits a suitable space \(L^ 0\) of measurable functions as its universal completion \(L^ u\). As no assumptions concerning the existence of order-continuous functionals on L are imposed, direct sums of (finite normal) submeasure spaces appear.
Our main objective is to study, by applying the representation result, minimal Hausdorff locally solid topologies on Riesz spaces. This is done in the second part of the paper, improving upon earlier results by Fremlin, Aliprantis and Burkinshaw. In particular, it is shown that a Riesz space L admits a minimal Hausdorff locally solid topology \(\lambda\) if and only if L admits the \(L^ 0\) obtained in the representation theorem as its universal completion and (modulo the embedding) \(\lambda\) is the restriction of the canonical topology of ”convergence in submeasure” on this \(L^ 0\).

MSC:

46A40	Ordered topological linear spaces, vector lattices
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
06F20	Ordered abelian groups, Riesz groups, ordered linear spaces

Keywords:

representation theorem; Archimedean Riesz space; universal completion; submeasure spaces; minimal Hausdorff locally solid topologies on Riesz spaces

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