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 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.

## Contents

* 1 Domain/Measure theory definition
* 1.1 Continuous valuation
* 1.2 Simple valuation
* 1.3 See also
* 2 Examples
* 2.1 Dirac valuation
* 3 See also
* 4 Notes
* 5 Works cited
* 6 External links

## Domain/Measure theory definition

Let (X,𝒯) be a topological space: a valuation is any set function v:𝒯->ℝ+∪{+∞} satisfying the following three properties v(∅)=0Strictness propertyv(U)≤v(V)ifU⊆VU,V∈𝒯Monotonicity propertyv(U∪V)+v(U∩V)=v(U)+v(V)∀U,V∈𝒯Modularity 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, Edalat & Saheb-Djahromi 2000 and Goubault-Larrecq 2005.

### Continuous valuation

A valuation (as defined in domain theory/measure theory) is said to be continuous if for every directed family {Ui}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 Ui⊆Uk and Uj⊆Uk) the following equality holds: v(⋃i∈IUi)=supi∈Iv(Ui).

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)=∑i=1naiδxi(U)∀U∈𝒯 where ai 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 vi(U)≤vk(U) and vj(U)≤vk(U)) is called quasi-simple valuation v¯(U)=supi∈Ivi(U)∀U∈𝒯.

### 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 Alvarez-Manilla, Edalat & Saheb-Djahromi 2000 and Goubault-Larrecq 2005 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.[lower-alpha 1]

## Examples

### Dirac valuation

Let (X,𝒯) be a topological space, and let x be a point of X: the map δx(U)={0ifx∉U1ifx∈U for all 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.

## See also

* Valuation (geometry)

## Notes

1. ↑ Details can be found in several arXiv papers of prof. Semyon Alesker.

## Works cited

* Alvarez-Manilla, Maurizio; Edalat, Abbas; Saheb-Djahromi, Nasser (2000), "An extension result for continuous valuations", Journal of the London Mathematical Society 61 (2): 629–640, doi:10.1112/S0024610700008681 .
* Goubault-Larrecq, Jean (2005), "Extensions of valuations", Mathematical Structures in Computer Science 15 (2): 271–297, doi:10.1017/S096012950400461X

## 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

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