Separating minimal valuations, point-continuous valuations and continuous valuations

Abstract

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 $\mu$ be the continuous valuation on $\mathcal{J}$ with $\mu (U)=1$ for nonempty Scott opens $U$ and $\mu (U)=0$ for $U=\varnothing$. Then $\mu$ is a point-continuous valuation on $\mathcal{J}$ that is not minimal. (2) Lebesgue measure extends to a measure on the Sorgenfrey line ${\mathbb{R}}_l$. Its restriction to the open subsets of ${\mathbb{R}}_l$ is a continuous valuation $\lambda$. Then its image valuation $\overline{\lambda }$ through the embedding of ${\mathbb{R}}_l$ into its Smyth powerdomain $\mathcal{Q}{\mathbb{R}}_l$ 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.