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 be a topological space: a valuation is any map
satisfying the following three properties
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 is said to be continuous if for every directed family of open sets the following equality holds:
This property is analogous to the τ-additivity of measures.

Simple valuation

A valuation is said to be simple if it is a finite linear combination with non-negative coefficients of Dirac valuations, i.e.
where is always greater than or at least equal to zero for all index. Simple valuations are obviously continuous in the above sense. The supremum of a directed family of [|simple valuations] is called quasi-simple valuation

Examples

Dirac valuation

Let be a topological space, and let ' be a point of ': the map
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.