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.

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

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 Alvarez-Manilla, Edalat & Saheb-Djahromi 2000 and Goubault-Larrecq 2005.

A valuation (as defined in domain theory/measure theory) is said to be continuous if for every directed family ${\displaystyle \{U_i{\}}_{i\in I}}$ of open sets (i.e. an indexed family of open sets which is also directed in the sense that for each pair of indexes ${\displaystyle i}$ and ${\displaystyle j}$ belonging to the index set ${\displaystyle I}$, there exists an index ${\displaystyle k}$ such that ${\displaystyle U_i\subseteq U_k}$ and ${\displaystyle U_j\subseteq U_k}$) the following equality holds: $${\displaystyle v\left(\bigcup _{i\in I}{U}_i\right)=\underset{i\in I}{\sup }v(U_i).}$$

This property is analogous to the τ-additivity of measures.

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, $${\displaystyle v(U)={\displaystyle \sum _{i=1}^n}{a}_i{\delta }_{x_i}(U)\mathrm{\forall }U\in {𝒯}}$$ where ${\displaystyle a_i}$ is always greater than or at least equal to zero for all index ${\displaystyle i}$. 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 ${\displaystyle i}$ and ${\displaystyle j}$ belonging to the index set ${\displaystyle I}$, there exists an index ${\displaystyle k}$ such that ${\displaystyle v_i(U)\le v_k(U)}$ and ${\displaystyle v_j(U)\le v_k(U)}$) is called quasi-simple valuation $${\displaystyle \overline{v}(U)=\underset{i\in I}{\sup }{v}_i(U)\mathrm{\forall }U\in {𝒯}.}$$

• 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]}

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

• Valuation (geometry)

• 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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