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 $(X,𝒯)$ be a topological space: a valuation is any set function $$v:𝒯\to {ℝ}^{+}\cup \{+\mathrm{\infty }\}$$ satisfying the following three properties $$\begin{array}{ccc}v(\varnothing )=0& & \text{Strictness property}\\ v(U)\le v(V)& \text{if}U\subseteq VU,V\in 𝒯& \text{Monotonicity property}\\ v(U\cup V)+v(U\cap V)=v(U)+v(V)& \mathrm{\forall }U,V\in 𝒯& \text{Modularity property}\end{array}$$

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 Template:Harvnb and Template:Harvnb.

Continuous valuation

A valuation (as defined in domain theory/measure theory) is said to be continuous if for every directed family $\{U_i{\}}_{i\in I}$ 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 $U_i\subseteq U_k$ and $U_j\subseteq U_k$) the following equality holds: $$v\left(\bigcup _{i\in I}{U}_i\right)=\underset{i\in I}{\sup }v(U_i).$$

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, $$v(U)=\sum _{i=1}^n{a}_i{\delta }_{x_i}(U)\mathrm{\forall }U\in 𝒯$$ 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 $v_i(U)\le v_k(U)$ and $v_j(U)\le v_k(U)$) is called quasi-simple valuation $$\overline{v}(U)=\underset{i\in I}{\sup }{v}_i(U)\mathrm{\forall }U\in 𝒯.$$

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 Template:Harvnb and Template:Harvnb 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.Template:Efn

Examples

Dirac valuation

Let $(X,𝒯)$ be a topological space, and let $x$ be a point of $X$: the map $${\delta }_x(U)=\{\begin{array}{cc}0& \text{if}x\notin U\\ 1& \text{if}x\in U\end{array}\text{ for all }U\in 𝒯$$ 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.

See also

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Works cited

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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