Probabilistic convergence transformation groups
-
T. M. G. Ahsanullah
und
Gunther Jäger
Abstract
We introduce a notion of a probabilistic convergence transformation group, and present various natural examples including quotient probabilistic convergence transformation group. In doing so, we construct a probabilistic convergence structure on the group of homeomorphisms and look into a probabilistic convergence group that arises from probabilistic uniform convergence structure on function spaces. Given a probabilistic convergence space, and an arbitrary group, we construct a probabilistic convergence transformation group. Introducing a notion of a probabilistic metric convergence transformation group on a probabilistic metric space, we obtain in a natural way a probabilistic convergence transformation group.
We gratefully acknowledge the support provided by the King Saud University, Deanship of Scientific Research, College of Science Research Center to carry out this work.
(Communicated by Anatolij Dvurečenskij)
Acknowledgement
We would like to thank the referees for giving their time to read our paper. We also express our gratitude to the Section Editor Professor Anatolij Dvurečenskij for his various remarks, and kind cooperation.
References
[1] Adámek, J.—Herrlich, H.—Strecker, G. E.: Abstract and Concrete Categories, J. Wiley & Sons, New York, 1990.Suche in Google Scholar
[2] Ahsanullah, T. M. G.—Jäger, G.: Probabilistic uniform convergence spaces redefined, Acta Math. Hungar. 146(2) (2015), 376–390.10.1007/s10474-015-0525-6Suche in Google Scholar
[3] Ahsanullah, T. M. G.—Jäger, G.: Probabilistic uniformization and probabilistic metrization of probabilistic convergence groups, Math. Slovaca 67(4) (2017), 985–1000.10.1515/ms-2017-0027Suche in Google Scholar
[4] Arens, R. F.: Topologies for homeomorphism groups, Amer. J. Math. 68 (1946), 593–610.10.2307/2371787Suche in Google Scholar
[5] Alsina, C.—Schweizer, B.—Sklar, A.: On the definition of a probabilistic normed space, Aequat. Math. 46 (1993), 91–98.10.1007/BF01834000Suche in Google Scholar
[6] Beattie, R.—Butzmann, H.-P.: Convergence Structures and Applications to Functional Analysis, Springer, 2002.10.1007/978-94-015-9942-9Suche in Google Scholar
[7] Brock, P.: Probabilistic convergence spaces and generalized metric spaces, Int. J. Math. Math. Sci. 21 (1998), 439–452.10.1155/S0161171298000611Suche in Google Scholar
[8] Brock, P.—Kent, D. C.: Approach spaces, limit tower spaces, and probabilistic convergence spaces, Appl. Categor. Struct. 5 (1997), 99–110.10.1023/A:1008633124960Suche in Google Scholar
[9] Borzová-Molnárová, J.—Halčinová, L.—Hutník, O.: Probabilistic-valued decomposable set functions with respect to triangle functions, Inform. Sci. 295 (2015), 347–357.10.1016/j.ins.2014.09.047Suche in Google Scholar
[10] Bourbaki, N.: Elements of Mathematics: General Topology, Part 2, Addison-Wesley Publishing Company, Reading, Massachusetts, 1966.Suche in Google Scholar
[11] Cook, C. H.—Fischer, H. R.: On equicontinuity and continuous convergence, Math. Ann. 159 (1965), 94–105.10.1007/BF01360283Suche in Google Scholar
[12] Cook, C. H.—Fischer, H. R.: Uniform convergence structures, Math. Ann. 173 (1967), 290–306.10.1007/BF01781969Suche in Google Scholar
[13] DieudonnÉ, J.: On topological groups of homeomorphisms, Amer. J. Math. 70(3) (1948), 659–680.10.2307/2372204Suche in Google Scholar
[14] Ellis, R.—Gottschalk, W. H.: Homomorphisms of transformation groups, Trans. Amer. Math. Soc. 94 (1960), 258–271.10.1090/S0002-9947-1960-0123635-1Suche in Google Scholar
[15] Florescu, L. C.: Probabilistic convergence structures, Aequat. Math. 38 (1989), 123–145.10.1007/BF01839999Suche in Google Scholar
[16] Ford, L. R.: Homeomorphism groups and coset spaces, Trans. Amer. Math. Soc. 77 (1954), 490–497.10.1090/S0002-9947-1954-0066636-1Suche in Google Scholar
[17] Fritsche, R.: Topologies for probabilistic metric spaces, Fund. Math. 72 (1971), 7–16.10.4064/fm-72-1-7-16Suche in Google Scholar
[18] Halčinová, L.—Hutník, O.: An integral with respect to probabilistic-valued decomposable measures, Internat. J. Approx. Reason. 55 (2014), 1469–1484.10.1016/j.ijar.2014.04.013Suche in Google Scholar
[19] Halčinová, L.—Hutník, O.—Mesiar, R.: On some classes of distance distritutive function-valued submeasures, Nonliear Analysis 74 (2011), 1545–1558.10.1016/j.na.2010.10.026Suche in Google Scholar
[20] Hewitt, E.—Ross, K. A.: Abstract Harmonic Analysis. Vol I: Structure of topological groups, Integration Theory, Group Representation, Springer-Verlag, Berlin, 1973.Suche in Google Scholar
[21] Hudson, S. N.: Transformation groups in the theory of topological loops, Proc. Amer. Math. Soc. 15 (1964), 872–877.10.1090/S0002-9939-1964-0167962-XSuche in Google Scholar
[22] Hutník, O.—Mesiar, R.: On a certain class of submeasures based on triangular norms, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 17 (2009), 297–316.10.1142/S0218488509005887Suche in Google Scholar
[23] Jäger, G.—Ahsanullah, T. M. G.: Probabilistic limit groups under a t-norm, Topology Proc. 44 (2014), 59–74.Suche in Google Scholar
[24] Jäger, G.: A convergence theory for probabilistic metric spaces, Quaest. Math. 38 (2015), 587–599.10.2989/16073606.2014.981734Suche in Google Scholar
[25] Lipovan, O.: Summeasures with probabilistic structures, Math. Morav. 4 (2000), 59–65.10.5937/MatMor0004059LSuche in Google Scholar
[26] Menger, K.: Statistical metrics, Proc. Nat. Acad. Sci. USA 28 (1942), 535–537.10.1007/978-3-7091-6045-9_35Suche in Google Scholar
[27] Montgomery, D.—Zippin, L.: Topological Transformation Groups, Interscience Publishers, Inc., New York, 1955.Suche in Google Scholar
[28] Nusser, H.: A generalization of probabilistic uniform spaces, Appl. Categor. Struct. 10 (2002), 81–98.10.1023/A:1013375301613Suche in Google Scholar
[29] Park, W. R.: Convergence structures on homeomorphism groups, Math. Ann. 199 (1972), 45–54.10.1007/BF01419575Suche in Google Scholar
[30] Pfanzagl, J.: Transformation groups and sufficient statistics, Ann. Math. Stat. 43 (1972), 553–568.10.1214/aoms/1177692636Suche in Google Scholar
[31] Poppe, H.: Compactness in General Function Spaces, VEB Deutscher Verlag der Wissenschaften, Berlin, 1974.Suche in Google Scholar
[32] Preuss, G.: Foundations of Topology: An Approach to Convenient Topology, Kluwer Academic Publishers, Dordrecht, 2002.10.1007/978-94-010-0489-3Suche in Google Scholar
[33] Rath, N.: Action of convergence groups, Topology Proc. 27(2) (2003), 601–1612.Suche in Google Scholar
[34] Richardson, G. D.—Kent, D. C.: Probabilistic convergence spaces, J. Austral. Math. Soc. 61 (1996), 400–420.10.1017/S1446788700000483Suche in Google Scholar
[35] Richardson, G. D.: Convergence in probabilistic semimetric spaces, Rocky Mountain J. Math. 18(3) (1988), 617–634.10.1216/RMJ-1988-18-3-617Suche in Google Scholar
[36] Saminger, S.—Sempi, C.: A primer on triangle functions I, Aequat. Math. 76 (2008), 201–240.10.1007/s00010-008-2936-8Suche in Google Scholar
[37] Schweizer, B.—Sklar, A.: Probabilistic Metric Spaces, North-Holland, New York, 1983.Suche in Google Scholar
[38] Sencimen, C.—Pehlivan, S.: Strong ideal convergence in probabilistic metric spaces, Proc. Indian Acad. Sci. 119(3) (2009), 401–410.10.1007/s12044-009-0028-xSuche in Google Scholar
[39] ŠErstnev, A. N.: On the notion of a random normed space, Dokl. Akad. Nauk SSSR 149 (1963) (in Russian), 280–283.Suche in Google Scholar
[40] Sherwood, H.: On E-spaces and their relation to other classes of probabilistic metric spaces, J. London Math. Soc. 44 (1969), 441–448.10.1112/jlms/s1-44.1.441Suche in Google Scholar
[41] Sibley, D. A.: A metric for weak convergence of distribution functions, Rocky Mountain J. Math. 1 (1971), 427–430.10.1216/RMJ-1971-1-3-427Suche in Google Scholar
[42] Tardiff, R. M.: Topologies for probabilistic metric spaces, Pacific J. Math. 65 (1976), 233–251.10.2140/pjm.1976.65.233Suche in Google Scholar
[43] Thorp, E.: Generalized topologies for statistical metric spaces, Fund. Math. 51 (1962), 9–12.10.4064/fm-51-1-9-21Suche in Google Scholar
[44] De Vries, J.: Topological Transformtion Groups 1. A Categorical Approach. Mathematical Centre Tracts 65, Mathematisch Centrum, Amsterdam, 1975.Suche in Google Scholar
[45] Wald, A.: On a statistical generalization of metric spaces, Proc. Nat. Acad. Sci. U. S. A. 29 (1943), 196–197.10.1073/pnas.29.6.196Suche in Google Scholar PubMed PubMed Central
[46] Yang, J. S.: Transformation groups of automorphisms of 𝒞(X, G), Proc. Amer. Math. Soc. 39 (1973), 619–624.10.2307/2039605Suche in Google Scholar
[47] Yang, J. S.: On isomorphic groups and homeomorphic spaces, Proc. Amer. Math. Soc. 41 (1974), 431–438.10.1090/S0002-9939-1974-0339060-XSuche in Google Scholar
© 2018 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Idempotents, group membership and their applications
- Residuation in non-associative MV-algebras
- An extension of F. Šik’s theorem on modular lattices
- Weak pseudo-BCK algebras
- Witt functor of a quadratic order
- A modification of a problem of Diophantus
- Cauchy problems involving a Hadamard-type fractional derivative
- Caratheodory’s solution of the Cauchy problem and a question of Z. Grande
- Sturm-Picone comparison theorems for nonlinear impulsive differential equations
- On oscillatory fourth order nonlinear neutral differential equations – III
- Local-periodic solutions for functional dynamic equations with infinite delay on changing-periodic time scales
- On the representation of involutive Jamesian functions
- Refinements of the heinz inequalities for operators and matrices
- Generalizations of Reid inequality
- Probabilistic convergence transformation groups
- Cluster sets and topology
- Some characterizations for Markov processes as mixed renewal processes
- Equivalent conditions of complete convergence for weighted sums of ANA random variables
Artikel in diesem Heft
- Idempotents, group membership and their applications
- Residuation in non-associative MV-algebras
- An extension of F. Šik’s theorem on modular lattices
- Weak pseudo-BCK algebras
- Witt functor of a quadratic order
- A modification of a problem of Diophantus
- Cauchy problems involving a Hadamard-type fractional derivative
- Caratheodory’s solution of the Cauchy problem and a question of Z. Grande
- Sturm-Picone comparison theorems for nonlinear impulsive differential equations
- On oscillatory fourth order nonlinear neutral differential equations – III
- Local-periodic solutions for functional dynamic equations with infinite delay on changing-periodic time scales
- On the representation of involutive Jamesian functions
- Refinements of the heinz inequalities for operators and matrices
- Generalizations of Reid inequality
- Probabilistic convergence transformation groups
- Cluster sets and topology
- Some characterizations for Markov processes as mixed renewal processes
- Equivalent conditions of complete convergence for weighted sums of ANA random variables
