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 $\mathbf{V}$ introduced by Jones and Plotkin, we show that the construction of continuous $R$-valuations extends to a strong monad $\mathbf{V}^R$ on the category of dcpos and Scott-continuous maps. Additionally, and as in recent work by the two authors and C. Th\'eron, and by the second author, B. Lindenhovius, M. Mislove and V. Zamdzhiev, we show that we can extract a commutative monad $\mathbf{V}^R_m$ 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 $\mathbf{I}\mathbb{R}^\star_+$, the interval domain of the extended nonnegative reals: (1) On every coherent topological space, every non-zero, bounded $\tau$-smooth measure $\mu$ (defined on the Borel $\sigma$-algebra), canonically determines a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation; and (2) such a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation is the most precise (in a certain sense) continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation that approximates $\mu$, when the support of $\mu$ 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 $\mathbf{I}\mathbb{R}^\star_+$-valuation. Additionally, we show that the latter is minimal.

翻译：本文在有向完备偏序集（简称dcpo）上引入连续R-赋值，作为domain理论中连续赋值的推广，将连续赋值的取值范围从实数扩展至所谓的阿贝尔d-环R。类似于Jones与Plotkin引入的赋值单子V，我们证明连续R-赋值的构造可扩展为dcpo与Scott连续映射范畴上的强单子V^R。此外，正如两位作者与C. Théron近期的工作，以及第二作者与B. Lindenhovius、M. Mislove、V. Zamdzhiev的研究所示，我们证明可从中提取出一个交换单子V^R_m，其元素称为最小R-赋值。我们还证明，当R取为Iℝ^⋆_+（即扩展非负实数的区间域）时，连续R-赋值与测度之间存在紧密联系：(1) 在每个凝聚拓扑空间上，任意非零有界τ-光滑测度μ（定义于Borel σ-代数上）典范地确定一个连续Iℝ^⋆_+-赋值；(2) 当μ的支撑集为第二可数稳定紧致拓扑空间的紧Hausdorff子空间时，这样的连续Iℝ^⋆_+-赋值在某种意义上是对μ最精确的逼近。这特别适用于单位区间上的Lebesgue测度，从而Lebesgue测度可被识别为一个连续Iℝ^⋆_+-赋值。进一步，我们证明后者是最小的。