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Separating minimal valuations, point-continuous valuations, and continuous valuations
Published online by Cambridge University Press: 07 December 2021
Abstract
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We give two concrete examples of continuous valuations on dcpo’s to separate minimal valuations, point-continuous valuations, and continuous valuations:
(1) Let
${\mathcal J}$ be the Johnstone’s non-sober dcpo, and μ be the continuous valuation on ${\mathcal J}$ with μ(U)=1 for nonempty Scott opens U and μ(U)=0 for 
$U=\emptyset$. Then, μ is a point-continuous valuation on ${\mathcal J}$ that is not minimal.(2) Lebesgue measure extends to a measure on the Sorgenfrey line
$\mathbb{R}_\ell$. Its restriction to the open subsets of $\mathbb{R}_\ell$ is a continuous valuation λ. Then, its image valuation 
$\overline\lambda$ through the embedding of $\mathbb{R}_\ell$ into its Smyth powerdomain 
$\mathcal{Q}\mathbb{R}_\ell$ in the Scott topology is a continuous valuation that is not point-continuous.
We believe that our construction $\overline\lambda$ might be useful in giving counterexamples displaying the failure of the general Fubini-type equations on dcpo’s.
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