Valuation (measure theory)

In measure theory or at least in the approach to it through 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 measure theory, probability theory and also in theoretical computer science.

Domain/Measure theory definition

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

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

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 Harvnb|Alvarez-Manilla|Edalat|Saheb-Djahromi|Nasser|2000 and Harvnb|Goulbault-Larrecq|2002.

Continuous valuation

A valuation (as defined in domain theory/measure theory) is said to be continuous if for "every directed family" $scriptstyle\; \{U\_i\}\_\{iin\; I\}$ "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 $scriptstyle\; U\_isubseteq\; U\_k$ and $scriptstyle\; U\_jsubseteq\; U\_k$) the following equality holds:

:

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.

:$v(U)=sum\_\{i=1\}^n\; a\_idelta\_\{x\_i\}(U)quadforall\; Uinmathcal\{T\}$

where $a\_i$ is always greather than or al 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 $scriptstyle\; v\_j(U)subseteq\; v\_k(U)!$) is called quasi-simple valuation

:

Related topics

* 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 Harvnb|Alvarez-Manilla|Edalat|Saheb-Djahromi|2000 and Harvnb|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 [http://arxiv.org/find/grp_q-bio,grp_cs,grp_physics,grp_math,grp_nlin/1/AND+au:+Alesker+ti:+Valuations/0/1/0/all/0/1 papers] of prof. Semyon Alesker.

Examples

Dirac valuation

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