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 $$ \scriptstyle (X,\mathcal{T})$$ be a topological space: a valuation is any set function $$v : \mathcal{T} \to \R^+ \cup \{+\infty\}$$ 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 and.

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: $$v\left(\bigcup_{i\in I}U_i\right) = \sup_{i\in I} v(U_i).$$

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, $$v(U)=\sum_{i=1}^n a_i\delta_{x_i}(U)\quad\forall U\in\mathcal{T}$$ 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 $$\scriptstyle v_i(U)\leq v_k(U)\!$$ and $$\scriptstyle v_j(U)\leq v_k(U)\!$$) is called quasi-simple valuation $$\bar{v}(U) = \sup_{i\in I}v_i(U) \quad \forall U\in \mathcal{T}.\,$$

Dirac valuation
Let $$ \scriptstyle (X,\mathcal{T})$$ be a topological space, and let $$x$$ be a point of $$X$$: the map $$\delta_x(U)= \begin{cases} 0 & \mbox{if}~x\notin U\\ 1 & \mbox{if}~x\in U \end{cases} \quad \text{ for all } U \in \mathcal{T} $$ 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.