Valuation (measure theory)

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In measure theory or at least in the approach to it through 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 measure theory, probability theory and also in theoretical computer science.

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[edit] 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.

[edit] 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).

[edit] 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 ai is always greather than or al 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}. \,

[edit] Related topics

  • 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 Goulbault-Larrecq 2002 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. Details can be found in several arxiv papers of prof. Seymon Alesker.

[edit] Examples

[edit] 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.