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 set function
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, that is,
where is nonnegative for all. 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.