Probabilistic metric space (original) (raw)
In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers R ≥ 0, but in distribution functions. Let D+ be the set of all probability distribution functions F such that F(0) = 0 (F is a nondecreasing, left continuous mapping from R into [0, 1] such that max(F) = 1). Then given a non-empty set S and a function F: S × S → D+ where we denote F(p, q) by Fp,q for every (p, q) ∈ S × S, the ordered pair (S, F) is said to be a probabilistic metric space if:
| Property | Value |
|---|---|
| dbo:abstract | In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers R ≥ 0, but in distribution functions. Let D+ be the set of all probability distribution functions F such that F(0) = 0 (F is a nondecreasing, left continuous mapping from R into [0, 1] such that max(F) = 1). Then given a non-empty set S and a function F: S × S → D+ where we denote F(p, q) by Fp,q for every (p, q) ∈ S × S, the ordered pair (S, F) is said to be a probabilistic metric space if: * For all u and v in S, u = v if and only if Fu,v(x) = 1 for all x > 0. * For all u and v in S, Fu,v = Fv,u. * For all u, v and w in S, Fu,v(x) = 1 and Fv,w(y) = 1 ⇒ Fu,w(x + y) = 1 for x, y > 0. (en) |
| dbo:thumbnail | wiki-commons:Special:FilePath/Probability_metric_DNN.png?width=300 |
| dbo:wikiPageID | 3891878 (xsd:integer) |
| dbo:wikiPageLength | 5068 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1088826402 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Euclidean_metric dbr:Standard_deviation dbr:Continuous_mapping dbr:Mathematics dbr:Maximum dbr:Mean dbr:Empty_set dbr:Identity_of_indiscernibles dbr:Ordered_pair dbc:Probability_distributions dbr:Distance dbr:Error_function dbr:Expected_value dbr:Normal_distribution dbr:Dirac_delta dbr:Probability_density_function dbr:Random_variable dbc:Metric_geometry dbr:Metric_space dbr:Real_numbers dbr:Metric_(mathematics) dbr:Random_vector dbr:File:Probability_metric_DNN.png |
| dbp:wikiPageUsesTemplate | dbt:Math dbt:Space dbt:Unreferenced dbt:Mathanalysis-stub |
| dcterms:subject | dbc:Probability_distributions dbc:Metric_geometry |
| gold:hypernym | dbr:Generalization |
| rdfs:comment | In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers R ≥ 0, but in distribution functions. Let D+ be the set of all probability distribution functions F such that F(0) = 0 (F is a nondecreasing, left continuous mapping from R into [0, 1] such that max(F) = 1). Then given a non-empty set S and a function F: S × S → D+ where we denote F(p, q) by Fp,q for every (p, q) ∈ S × S, the ordered pair (S, F) is said to be a probabilistic metric space if: (en) |
| rdfs:label | Probabilistic metric space (en) |
| owl:sameAs | freebase:Probabilistic metric space yago-res:Probabilistic metric space wikidata:Probabilistic metric space https://global.dbpedia.org/id/4tYbB |
| prov:wasDerivedFrom | wikipedia-en:Probabilistic_metric_space?oldid=1088826402&ns=0 |
| foaf:depiction | wiki-commons:Special:FilePath/Probability_metric_DNN.png |
| foaf:isPrimaryTopicOf | wikipedia-en:Probabilistic_metric_space |
| is dbo:wikiPageWikiLink of | dbr:Approach_space dbr:Statistical_distance dbr:T-norm dbr:List_of_statistics_articles |
| is foaf:primaryTopic of | wikipedia-en:Probabilistic_metric_space |
