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 ${\displaystyle \scriptstyle (X,{\mathcal {T}})}$ be a topological space: a valuation is any map

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

satisfying the following three properties

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

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 ${\displaystyle \scriptstyle \{U_{i}\}_{i\in I}}$ of open sets (i.e. an indexed family of open sets which is also directed in the sense that for each pair of indexes ${\displaystyle i}$ and ${\displaystyle j}$ belonging to the index set ${\displaystyle I}$, there exists an index ${\displaystyle k}$ such that ${\displaystyle \scriptstyle U_{i}\subseteq U_{k}}$ and ${\displaystyle \scriptstyle U_{j}\subseteq U_{k}}$) the following equality holds:

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

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, i.e.

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

where ${\displaystyle a_{i}}$ is always greater than or at least equal to zero for all index ${\displaystyle i}$. 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 ${\displaystyle i}$ and ${\displaystyle j}$ belonging to the index set ${\displaystyle I}$, there exists an index ${\displaystyle k}$ such that ${\displaystyle \scriptstyle v_{i}(U)\leq v_{k}(U)\!}$ and ${\displaystyle \scriptstyle v_{j}(U)\leq v_{k}(U)\!}$) is called quasi-simple valuation

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

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.[lower-alpha 1]

Examples

Dirac valuation

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

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

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.

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, 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