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 valuationedit
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 alsoedit
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 alsoedit
Valuation (geometry) – in geometryPages displaying wikidata descriptions as a fallback
Notesedit
^Details can be found in several arXiv papers of prof. Semyon Alesker.
Works citededit
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
External linksedit
Alesker, Semyon, "various preprints on valuation s", arXiv preprint server, primary site at Cornell University. Several papers dealing with valuations on convex sets, valuations on manifolds and related topics.
The nLab page on valuations
December 01, 2023
valuation, measure, theory, measure, theory, least, approach, domain, theory, valuation, from, class, open, sets, topological, space, positive, real, numbers, including, infinity, with, certain, properties, concept, closely, related, that, measure, such, finds. 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 Contents 1 Domain Measure theory definition 1 1 Continuous valuation 1 2 Simple valuation 1 3 See also 2 Examples 2 1 Dirac valuation 3 See also 4 Notes 5 Works cited 6 External linksDomain Measure theory definition editLet X T displaystyle scriptstyle X mathcal T nbsp be a topological space a valuation is any set functionv T R displaystyle v mathcal T to mathbb R cup infty nbsp satisfying the following three properties v 0 Strictness property v U v V if U V U V T Monotonicity property v U V v U V v U v V U V T Modularity property displaystyle begin array lll v varnothing 0 amp amp scriptstyle text Strictness property v U leq v V amp mbox if U subseteq V quad U V in mathcal T amp scriptstyle text Monotonicity property v U cup V v U cap V v U v V amp forall U V in mathcal T amp scriptstyle text Modularity property end array nbsp 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 amp Saheb Djahromi 2000 and Goubault Larrecq 2005 Continuous valuation edit A valuation as defined in domain theory measure theory is said to be continuous if for every directed family U i i I displaystyle scriptstyle U i i in I nbsp 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 displaystyle i nbsp and j displaystyle j nbsp belonging to the index set I displaystyle I nbsp there exists an index k displaystyle k nbsp such that U i U k displaystyle scriptstyle U i subseteq U k nbsp and U j U k displaystyle scriptstyle U j subseteq U k nbsp the following equality holds v i I U i sup i I v U i displaystyle v left bigcup i in I U i right sup i in I v U i nbsp This property is analogous to the t additivity of measures Simple valuation edit 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 i 1 n a i d x i U U T displaystyle v U sum i 1 n a i delta x i U quad forall U in mathcal T nbsp where a i displaystyle a i nbsp is always greater than or at least equal to zero for all index i displaystyle i nbsp 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 displaystyle i nbsp and j displaystyle j nbsp belonging to the index set I displaystyle I nbsp there exists an index k displaystyle k nbsp such that v i U v k U displaystyle scriptstyle v i U leq v k U nbsp and v j U v k U displaystyle scriptstyle v j U leq v k U nbsp is called quasi simple valuation v U sup i I v i U U T displaystyle bar v U sup i in I v i U quad forall U in mathcal T nbsp See also edit 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 amp 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 a Examples editDirac valuation edit Let X T displaystyle scriptstyle X mathcal T nbsp be a topological space and let x displaystyle x nbsp be a point of X displaystyle X nbsp the mapd x U 0 if x U 1 if x U for all U T displaystyle delta x U begin cases 0 amp mbox if x notin U 1 amp mbox if x in U end cases quad text for all U in mathcal T nbsp 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 editValuation geometry in geometryPages displaying wikidata descriptions as a fallbackNotes edit Details can be found in several arXiv papers of prof Semyon Alesker Works cited editAlvarez Manilla Maurizio Edalat Abbas Saheb Djahromi Nasser 2000 An extension result for continuous valuations Journal of the London Mathematical Society 61 2 629 640 CiteSeerX 10 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 S096012950400461XExternal links editAlesker Semyon various preprints on valuation s arXiv preprint server primary site at Cornell University Several papers dealing with valuations on convex sets valuations on manifolds and related topics The nLab page on valuations Retrieved from https en wikipedia org w index php title Valuation measure theory amp oldid 1095513200, wikipedia, wiki, book, books, library,