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
Alvarez-Manilla, Edalat & Saheb-Djahromi 2000 and
Goubault-Larrecq 2005.
Continuous valuation
A valuation (as defined in domain theory/measure theory) is said to be continuous if for every directed familyof
open sets (i.e. an
indexed family of open sets which is also
directed in the sense that for each pair of indexes and belonging to the
index set, there exists an index such that and ) the following
equality holds:
This property is analogous to the
τ-additivity of measures.
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 (i.e. an indexed family of simple valuations which is also directed in the sense that for each pair of indexes and belonging to the index set , there exists an index such that and ) is called quasi-simple valuation
See also
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.
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.
See also
Valuation (geometry) – in geometryPages displaying wikidata descriptions as a fallback
Notes
^Details can be found in several
arXivpapers of prof. Semyon Alesker.
Works cited
Alvarez-Manilla, Maurizio; Edalat, Abbas; Saheb-Djahromi, Nasser (2000), "An extension result for continuous valuations", Journal of the London Mathematical Society, 61 (2): 629–640,
CiteSeerX10.1.1.23.9676,
doi:
10.1112/S0024610700008681.
Goubault-Larrecq, Jean (2005), "Extensions of valuations", Mathematical Structures in Computer Science, 15 (2): 271–297,
doi:
10.1017/S096012950400461X