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.