No abstract
We study G δ subspaces of continuous dcpos, which we call domaincomplete spaces, and G δ subspaces of locally compact sober spaces, which we call LCS-complete spaces. Those include all locally compact sober spaces-in particular, all continuous dcpos-, all topologically complete spaces in the sense ofČech, and all quasi-Polish spaces-in particular, all Polish spaces. We show that LCS-complete spaces are sober, Wilker, compactly Choquet-complete, completely Baire, and -consonant-in particular, consonant; that the countably-based LCS-complete (resp., domaincomplete) spaces are the quasi-Polish spaces exactly; and that the metrizable LCS-complete (resp., domain-complete) spaces are the completely metrizable spaces. We include two applications: on LCS-complete spaces, all continuous valuations extend to measures, and sublinear previsions form a space homeomorphic to the convex Hoare powerdomain of the space of continuous valuations.Let us start with the following question: for which class of topological spaces X is it true that every (locally finite) continuous valuation on X extends to a measure on X, with its Borel σ-algebra? The question is well-studied, and Klaus Keimel and Jimmie Lawson have rounded it up nicely in [24]. A result by Mauricio Alvarez-Manilla et al.[2] (see also Theorem 5.3 of the paper by Keimel and Lawson) states that every locally compact sober space fits.Locally compact sober spaces are a pretty large class of spaces, including many non-Hausdorff spaces, and in particular all the continuous dcpos of domain theory. However, such a result will be of limited use to the ordinary measure theorist, who is used to working with Polish spaces, including such spaces as Baire space N N , which is definitely not a locally compact space.It is not too hard to extend the above theorem to the following larger class of spaces (and to drop the local finiteness assumption as well):Theorem 1.1 Let X be a (homeomorph of a) G δ subset of a locally compact sober space Y . Every continuous valuation ν on X extends to a measure on X with its Borel σ-algebra.We defer the proof of that result to Section 18. The point is that we do have a measure extension theorem on a class of spaces that contains both the continuous dcpos of domain theory and the Polish spaces of topological measure theory. We will call such spaces LCS-complete, and we are aware that this is probably not an optimal name. Topologically complete would have been a better name, if it had not been taken already [5].Another remarkable class of spaces is the class of quasi-Polish spaces, discovered and studied by the first author [7]. This one generalizes both ω-continuous dcpos and Polish spaces, and we will see in Section 5 that the class of LCScomplete spaces is a proper superclass. We will also see that there is no countably-based LCS-complete space that would fail to be quasi-Polish. Hence LCS-complete spaces can be seen as an extension of the notion of quasi-Polish spaces, and the extension is strict only for non-countably based spaces.Generally, our pur...
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