In geometry, a valuation is a finitely additive function on a collection of admissible subsets of a fixed set with values in an abelian semigroup. For example, the Lebesgue measure is a valuation on finite unions of convex bodies (i.e. non-empty compact convex sets) of Euclidean space . Other examples of valuations on finite unions of convex bodies are the surface area, the mean width, and the Euler characteristic. In the geometric setting, often continuity (or smoothness) conditions are imposed on valuations, but there are also purely discrete facets of the theory. In fact, the concept of valuation has its origin in the dissection theory of polytopes and in particular Hilbert's third problem, which has grown into a rich theory, heavily reliant on advanced tools from abstract algebra.