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 [math]\displaystyle{ \scriptstyle (X,\mathcal{T}) }[/math] be a topological space: a valuation is any set function [math]\displaystyle{ v : \mathcal{T} \to \R^+ \cup \{+\infty\} }[/math] satisfying the following three properties [math]\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} }[/math]
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 [math]\displaystyle{ \scriptstyle \{U_i\}_{i\in I} }[/math] of open sets (i.e. an indexed family of open sets which is also directed in the sense that for each pair of indexes [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math] belonging to the index set [math]\displaystyle{ I }[/math], there exists an index [math]\displaystyle{ k }[/math] such that [math]\displaystyle{ \scriptstyle U_i\subseteq U_k }[/math] and [math]\displaystyle{ \scriptstyle U_j\subseteq U_k }[/math]) the following equality holds: [math]\displaystyle{ v\left(\bigcup_{i\in I}U_i\right) = \sup_{i\in I} v(U_i). }[/math]
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, [math]\displaystyle{ v(U)=\sum_{i=1}^n a_i\delta_{x_i}(U)\quad\forall U\in\mathcal{T} }[/math] where [math]\displaystyle{ a_i }[/math] is always greater than or at least equal to zero for all index [math]\displaystyle{ i }[/math]. 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 [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math] belonging to the index set [math]\displaystyle{ I }[/math], there exists an index [math]\displaystyle{ k }[/math] such that [math]\displaystyle{ \scriptstyle v_i(U)\leq v_k(U)\! }[/math] and [math]\displaystyle{ \scriptstyle v_j(U)\leq v_k(U)\! }[/math]) is called quasi-simple valuation [math]\displaystyle{ \bar{v}(U) = \sup_{i\in I}v_i(U) \quad \forall U\in \mathcal{T}.\, }[/math]
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 [math]\displaystyle{ \scriptstyle (X,\mathcal{T}) }[/math] be a topological space, and let [math]\displaystyle{ x }[/math] be a point of [math]\displaystyle{ X }[/math]: the map [math]\displaystyle{ \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} }[/math] 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
Notes
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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