Valuation (measure theory)
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 be a topological space: a valuation is any map
satisfying the following three properties
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 of open sets (i.e. an indexed family of open sets which is also directed in the sense that for each pair of indexes and belonging to the index set , there exists an index such that and ) the following equality holds:
where is always greather than or at least equal to zero for all index . 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 and belonging to the index set , there exists an index such that and ) is called quasi-simple valuation
- 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 be a topological space, and let be a point of : the map
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.
- Alvarez-Manilla, Maurizio; Edalat, Abbas; Saheb-Djahromi, Nasser (2000), "An extension result for continuous valuations", Journal of the London Mathematical Society 61: 629–640, doi:10.1112/S0024610700008681. DOI:10.1112/S0024610700008681. The preprint from the homepage of the second author is freely readable.
- Goulbault-Larrecq, Jean (November 2002), "Extensions of valuations", Research Report LSV-02-17 (France: LSV (Laboratoire de Spécification at Vérification), CNRS & ENS de Cachan). Published as "Extension of valuations", Mathematical Structures in Computer Science (2005), 15: 271-297, DOI:10.1017/S096012950400461X.