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 positive real numbers including infinity. 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 map

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

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 et al. 2000 and Goulbault-Larrecq 2002.

Continuous valuation

A valuation (as defined in domain theory/measure theory) is said to be continuous if for every directed family $\scriptstyle \{U_i\}_{i\in I}$ 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:

$v\left(\bigcup_{i\in I}U_i\right) = \sup_{i\in I} v(U_i).$

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.

$v(U)=\sum_{i=1}^n a_i\delta_{x_i}(U)\quad\forall U\in\mathcal{T}$

where $a_i$ is always greather 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 $\scriptstyle v_j(U)\subseteq v_k(U)\!$) 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 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 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

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

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.

References

• Alvarez-Manilla, Maurizio; Edalat, Abbas; Saheb-Djahromi, Nasser (2000), "An extension result for continuous valuations", Journal of the London Mathematical Society 61: 629–640, doi:10.1112/S0024610700008681. DOI:10.1112/S0024610700008681. The preprint from the homepage of the second author is freely readable.
• Goulbault-Larrecq, Jean (November 2002), "Extensions of valuations", Research Report LSV-02-17 (France: LSV (Laboratoire de Spécification at Vérification), CNRS & ENS de Cachan). Published as "Extension of valuations", Mathematical Structures in Computer Science (2005), 15: 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.