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 π introduced by Jones and Plotkin, we show that the construction of continuous R-valuations extends to a strong monad π^R on the category of dcpos and Scott-continuous maps. Additionally, and as in recent work by the two authors and C. ThΓ©ron, and by the second author, B. Lindenhovius, M. Mislove and V. Zamdzhiev, we show that we can extract a commutative monad π^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 πβ^β_+, the interval domain of the extended nonnegative reals: (1) On every coherent topological space, every non-zero, bounded Ο-smooth measure ΞΌ (defined on the Borel Ο-algebra), canonically determines a continuous πβ^β_+-valuation; and (2) such a continuous πβ^β_+-valuation is the most precise (in a certain sense) continuous πβ^β_+-valuation that approximates ΞΌ, when the support of ΞΌ 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 πβ^β_+-valuation. Additionally, we show that the latter is minimal.
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