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