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 drags 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 VR 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 VRm 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.