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}}.\,

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 External link in |publisher= (help). 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.

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