Separating minimal valuations, point-continuous valuations and continuous valuations
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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 𝒥 be the Johnstone's non-sober dcpo, and μ be the continuous valuation on 𝒥 with μ(U) =1 for nonempty Scott opens U and μ(U) = 0 for U=∅. Then μ is a point-continuous valuation on 𝒥 that is not minimal. (2) Lebesgue measure extends to a measure on the Sorgenfrey line ℝ_l. Its restriction to the open subsets of ℝ_l is a continuous valuation λ. Then its image valuation λ through the embedding of ℝ_l into its Smyth powerdomain 𝒬ℝ_l in the Scott topology is a continuous valuation that is not point-continuous. We believe that our construction λ might be useful in giving counterexamples displaying the failure of the general Fubini-type equations on dcpo's.
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