Valuation (measure theory)

Let $\backslash scriptstyle\; (X,\backslash mathcal\{T\})$ be a topological space: a valuation is any set function

$$v\; :\; \backslash mathcal\{T\}\; \backslash to\; \backslash R^+\; \backslash cup\; \backslash \{+\backslash infty\backslash \}$$

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 {{Harvnb|Alvarez-Manilla|Edalat|Saheb-Djahromi|2000}} and {{Harvnb|Goubault-Larrecq|2005}}.

A valuation (as defined in domain theory/measure theory) is said to be continuous if for every directed family $\backslash scriptstyle\; \backslash \{U\_i\backslash \}\_\{i\backslash in\; I\}$ 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 $\backslash scriptstyle\; U\_i\backslash subseteq\; U\_k$ and $\backslash scriptstyle\; U\_j\backslash subseteq\; U\_k$) the following equality holds:

$$v\backslash left(\backslash bigcup\_\{i\backslash in\; I\}U\_i\backslash right)\; =\; \backslash sup\_\{i\backslash in\; I\}\; v(U\_i).$$

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,

$$v(U)=\backslash sum\_\{i=1\}^n\; a\_i\backslash delta\_\{x\_i\}(U)\backslash quad\backslash forall\; U\backslash in\backslash mathcal\{T\}$$

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 $\backslash scriptstyle\; v\_i(U)\backslash leq\; v\_k(U)\backslash !$ and $\backslash scriptstyle\; v\_j(U)\backslash leq\; v\_k(U)\backslash !$) is called quasi-simple valuation

$$\backslash bar\{v\}(U)\; =\; \backslash sup\_\{i\backslash in\; I\}v\_i(U)\; \backslash quad\; \backslash forall\; U\backslash in\; \backslash mathcal\{T\}.\backslash ,$$

• 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 {{Harvnb|Alvarez-Manilla|Edalat|Saheb-Djahromi|2000}} and {{Harvnb|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.{{efn|Details can be found in several arXiv [https://arxiv.org/find/grp_q-bio,grp_cs,grp_physics,grp_math,grp_nlin/1/AND+au:+Alesker+ti:+Valuations/0/1/0/all/0/1 papers] of prof. Semyon Alesker.}}

Let $\backslash scriptstyle\; (X,\backslash mathcal\{T\})$ be a topological space, and let $x$ be a point of $X$: the map

$$\backslash delta\_x(U)=$$

\begin{cases}

0 & \mbox{if}~x\notin U\\

1 & \mbox{if}~x\in U

\end{cases}

\quad \text{ for all } 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.

• {{annotated link|Valuation (geometry)}}

{{notelist}}

{{refbegin}}

• {{Citation| title = An extension result for continuous valuations

| last1 = Alvarez-Manilla | first1 = Maurizio

| last2 = Edalat | first2 = Abbas

| last3 = Saheb-Djahromi | first3 = Nasser

| journal = Journal of the London Mathematical Society

| year = 2000 | volume = 61 | issue = 2 | pages = 629–640

| citeseerx = 10.1.1.23.9676 | doi = 10.1112/S0024610700008681

}}.

• {{Citation| title = Extensions of valuations

| last = Goubault-Larrecq | first = Jean | year = 2005

| journal = Mathematical Structures in Computer Science

| volume = 15 | issue = 2 | pages = 271–297

| doi = 10.1017/S096012950400461X

}}

{{refend}}

• Alesker, Semyon, "[https://arxiv.org/find/grp_q-bio,grp_cs,grp_physics,grp_math,grp_nlin/1/AND+au:+Alesker+ti:+Valuations/0/1/0/all/0/1 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.
• [https://ncatlab.org/nlab/show/valuation+%28measure+theory%29 The nLab page on valuations]

{{Measure theory}}

Category:Measure theory