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 $\scriptstyle (X,\mathcal)$ be a topological space: a valuation is any set function
$v : \mathcal \to \R^+ \cup \$
satisfying the following three properties
$\begin v(\varnothing) = 0 & & \scriptstyle\\ v(U)\leq v(V) & \mbox~U\subseteq V\quad U,V\in\mathcal & \scriptstyle\\ v(U\cup V)+ v(U\cap V) = v(U)+v(V) & \forall U,V\in\mathcal & \scriptstyle\, \end$

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 and .

Continuous valuation

A valuation (as defined in domain theory/measure theory) is said to be continuous if for ''every directed family'' $\scriptstyle \_$ ''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:
$v\left(\bigcup_U_i\right) = \sup_ v(U_i).$

This property is analogous to the Ï„-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,
$v(U)=\sum_^n a_i\delta_(U)\quad\forall U\in\mathcal$
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
$\bar(U) = \sup_v_i(U) \quad \forall U\in \mathcal.\,$

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 and 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 $\scriptstyle (X,\mathcal)$ be a topological space, and let ''$x$'' be a point of ''$X$'': the map
$\delta_x(U)= \begin 0 & \mbox~x\notin U\\ 1 & \mbox~x\in U \end \quad \text U \in \mathcal$
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

*.
*

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

{{Measure theory

Measure theory