اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSeparating minimal valuations, point-continuous valuations and continuous valuations
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We give two concrete examples of continuous valuations on dcpos to separate minimal valuations, point-continuous valuations and continuous valuations: (1) Let $mathcal J$ be the Johnstones 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 $U=emptyset$. 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 $overlinelambda$ through the embedding of $mathbb R_{l}$ into its Smyth powerdomain $mathcal Qmathbb R_{l}$ in the Scott topology is a continuous valuation that is not point-continuous. We believe that our construction $overlinelambda$ might be useful in giving counterexamples displaying the failure of the general Fubini-type equations on dcpos.
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