Valuation (measure theory)

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

$${\displaystyle v\left(\bigcup _{i\in I}U_{i}\right)=\sup _{i\in I}v(U_{i}).}$$

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

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,

$${\displaystyle v(U)=\sum _{i=1}^{n}a_{i}\delta _{x_{i}}(U)\quad \forall U\in {\mathcal {T}}}$$

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 ${\displaystyle \scriptstyle v_{i}(U)\leq v_{k}(U)\!}$ and ${\displaystyle \scriptstyle v_{j}(U)\leq v_{k}(U)\!}$) is called quasi-simple valuation

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

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

$${\displaystyle \delta _{x}(U)={\begin{cases}0&{\mbox{if}}~x\notin U\\1&{\mbox{if}}~x\in U\end{cases}}\quad {\text{ for all }}U\in {\mathcal {T}}}$$

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.