Valuation (measure theory) explained

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 of positive real numbers including infinity, with certain properties. 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

\scriptstyle (X, l{T}) be a topological space: a valuation is any set function$$v\; :\; \backslash mathcal\; \backslash to\; \backslash R^+\; \backslash cup\; \backslash$$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 and .

Continuous valuation

A valuation (as defined in domain theory/measure theory) is said to be continuous if for every directed family

\scriptstyle \{U _{i\}} _{i\in} 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 \scriptstyle U _{i\subseteq} U _{k} and \scriptstyle U _{j\subseteq} U _{k} ) the following equality holds:$$v\backslash left(\backslash bigcup\_U\_i\backslash right)\; =\; \backslash sup\_\; v(U\_i).$$

This property is analogous to the τ-additivity of measures.

Simple valuation

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, that is,$$v(U)=\backslash sum\_^n\; a\_i\backslash delta\_(U)\backslash quad\backslash forall\; U\backslash in\backslash mathcal$$where

a _{i} is always greater 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 \scriptstyle v _{i(U)\leq} v _{k(U)} and \scriptstyle v _{j(U)\leq} v _{k(U)} ) is called quasi-simple valuation$$\backslash bar(U)\; =\; \backslash sup\_v\_i(U)\; \backslash quad\; \backslash forall\; U\backslash in\; \backslash mathcal.\backslash ,$$

See also

• 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 and 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.

Examples

Dirac valuation

Let

\scriptstyle (X, l{T}) be a topological space, and let x be a point of X : the mapis 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.

Works cited

• .

External links

• Alesker, Semyon, "various preprints on valuation s", arXiv preprint server, primary site at Cornell University. Several papers dealing with valuations on convex sets, valuations on manifolds and related topics.
• The nLab page on valuations