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.

Domain/Measure theory definition

Let $\scriptstyle (X,{\mathcal {T}})$ be a topological space: a valuation is any set function

$$v:{\mathcal {T}}\to \mathbb {R} ^{+}\cup \{+\infty \}$$

satisfying the following three properties

$${\begin{array}{lll}v(\varnothing )=0&&\scriptstyle {\text{Strictness property}}\\v(U)\leq v(V)&{\mbox{if}}~U\subseteq V\quad U,V\in {\mathcal {T}}&\scriptstyle {\text{Monotonicity property}}\\v(U\cup V)+v(U\cap V)=v(U)+v(V)&\forall U,V\in {\mathcal {T}}&\scriptstyle {\text{Modularity property}}\,\end{array}}$$

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 $\scriptstyle \{U_{i}\}_{i\in I}$ 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 $\scriptstyle U_{i}\subseteq U_{k}$ and $\scriptstyle U_{j}\subseteq U_{k}$) the following equality holds:

$$v\left(\bigcup _{i\in I}U_{i}\right)=\sup _{i\in I}v(U_{i}).$$

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,

$$v(U)=\sum _{i=1}^{n}a_{i}\delta _{x_{i}}(U)\quad \forall U\in {\mathcal {T}}$$

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

$${\bar {v}}(U)=\sup _{i\in I}v_{i}(U)\quad \forall U\in {\mathcal {T}}.\,$$

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.

Examples

Dirac valuation

Let $\scriptstyle (X,{\mathcal {T}})$ be a topological space, and let $x$ be a point of $X$: the map

$$\delta _{x}(U)={\begin{cases}0&{\mbox{if}}~x\notin U\\1&{\mbox{if}}~x\in U\end{cases}}\quad {\text{ for all }}U\in {\mathcal {T}}$$

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) – in geometryPages displaying wikidata descriptions as a fallback

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, CiteSeerX 10.1.1.23.9676, 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