In measure theory, or at least in the approach to it via the domain theory, a valuation is a map from the class of open sets of a topological space to the set positive real numbers including infinity. It is a concept closely related to that of a measure, and as such, it finds applications in measure theory, probability theory, and theoretical computer science.
Domain/Measure theory definition
Let ( X , T ) be a topological space: a valuation is any map
v : T → R + ∪ { + ∞ } satisfying the following three properties
v ( ∅ ) = 0 Strictness property v ( U ) ≤ v ( V ) if U ⊆ V U , V ∈ T Monotonicity property v ( U ∪ V ) + v ( U ∩ V ) = v ( U ) + v ( V ) ∀ U , V ∈ T Modularity property The definition immediately shows the relationship between a valuation and a measure: the properties of the two mathematical object are often very similar if not identical, the only difference being that the domain of a measure is the Borel algebra of the given topological space, while the domain of a valuation is the class of open sets. Further details and references can be found in Alvarez-Manilla et al. 2000 and Goulbault-Larrecq 2002.
A valuation (as defined in domain theory/measure theory) is said to be continuous if for every directed family { U i } i ∈ I of open sets (i.e. an indexed family of open sets which is also directed in the sense that for each pair of indexes i and j belonging to the index set I , there exists an index k such that U i ⊆ U k and U j ⊆ U k ) the following equality holds:
v ( ⋃ i ∈ I U i ) = sup i ∈ I v ( U i ) . A valuation (as defined in domain theory/measure theory) is said to be simple if it is a finite linear combination with non-negative coefficients of Dirac valuations, i.e.
v ( U ) = ∑ i = 1 n a i δ x i ( U ) ∀ U ∈ T where a i is always greather than or at least equal to zero for all index i . Simple valuations are obviously continuous in the above sense. The supremum of a directed family of simple valuations (i.e. an indexed family of simple valuations which is also directed in the sense that for each pair of indexes i and j belonging to the index set I , there exists an index k such that v i ( U ) ≤ v k ( U ) and v j ( U ) ⊆ v k ( U ) ) is called quasi-simple valuation
v ¯ ( U ) = sup i ∈ I v i ( U ) ∀ U ∈ T . The extension problem for a given valuation (in the sense of domain theory/measure theory) consists in finding under what type of conditions it can be extended to a measure on a proper topological space, which may or may not be the same space where it is defined: the papers Alvarez-Manilla, Edalat & Saheb-Djahromi 2000 and Goulbault-Larrecq 2002 in the reference section are devoted to this aim and give also several historical details.The concepts of valuation on convex sets and valuation on manifolds are a generalization of valuation in the sense of domain/measure theory. A valuation on convex sets is allowed to assume complex values, and the underlying topological space is the set of non-empty convex compact subsets of a finite-dimensional vector space: a valuation on manifolds is a complex valued finitely additive measure defined on a proper subset of the class of all compact submanifolds of the given manifolds. Details can be found in several arxiv papers of prof. Semyon Alesker.Let ( X , T ) be a topological space, and let x be a point of X : the map
δ x ( U ) = { 0 if x ∉ U 1 if x ∈ U ∀ U ∈ T is a valuation in the domain theory/measure theory, sense called Dirac valuation. This concept bears its origin from distribution theory as it is an obvious transposition to valuation theory of Dirac distribution: as seen above, Dirac valuations are the "bricks" simple valuations are made of.
