Continuous R-valuations
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We introduce continuous R-valuations on directed-complete posets (dcpos, for short), as a generalization of continuous valuations in domain theory, by extending values of continuous valuations from reals to so-called Abelian d-rags R. Like the valuation monad 𝐕 introduced by Jones and Plotkin, we show that the construction of continuous R-valuations extends to a strong monad 𝐕^R on the category of dcpos and Scott-continuous maps. Additionally, and as in recent work by the two authors and C. Théron, and by the second author, B. Lindenhovius, M. Mislove and V. Zamdzhiev, we show that we can extract a commutative monad 𝐕^R_m out of it, whose elements we call minimal R-valuations. We also show that continuous R-valuations have close connections to measures when R is taken to be 𝐈ℝ^⋆_+, the interval domain of the extended nonnegative reals: (1) On every coherent topological space, every non-zero, bounded τ-smooth measure μ (defined on the Borel σ-algebra), canonically determines a continuous 𝐈ℝ^⋆_+-valuation; and (2) such a continuous 𝐈ℝ^⋆_+-valuation is the most precise (in a certain sense) continuous 𝐈ℝ^⋆_+-valuation that approximates μ, when the support of μ is a compact Hausdorff subspace of a second-countable stably compact topological space. This in particular applies to Lebesgue measure on the unit interval. As a result, the Lebesgue measure can be identified as a continuous 𝐈ℝ^⋆_+-valuation. Additionally, we show that the latter is minimal.
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