Probabilistic convergence transformation groups
-
T. M. G. Ahsanullah
and
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.Search 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/s10474-015-0525-6Search 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-2017-0027Search in Google Scholar
[4] Arens, R. F.: Topologies for homeomorphism groups, Amer. J. Math. 68 (1946), 593–610.10.2307/2371787Search 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/BF01834000Search in Google Scholar
[6] Beattie, R.—Butzmann, H.-P.: Convergence Structures and Applications to Functional Analysis, Springer, 2002.10.1007/978-94-015-9942-9Search in Google Scholar
[7] Brock, P.: Probabilistic convergence spaces and generalized metric spaces, Int. J. Math. Math. Sci. 21 (1998), 439–452.10.1155/S0161171298000611Search 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:1008633124960Search 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.2014.09.047Search in Google Scholar
[10] Bourbaki, N.: Elements of Mathematics: General Topology, Part 2, Addison-Wesley Publishing Company, Reading, Massachusetts, 1966.Search in Google Scholar
[11] Cook, C. H.—Fischer, H. R.: On equicontinuity and continuous convergence, Math. Ann. 159 (1965), 94–105.10.1007/BF01360283Search in Google Scholar
[12] Cook, C. H.—Fischer, H. R.: Uniform convergence structures, Math. Ann. 173 (1967), 290–306.10.1007/BF01781969Search in Google Scholar
[13] DieudonnÉ, J.: On topological groups of homeomorphisms, Amer. J. Math. 70(3) (1948), 659–680.10.2307/2372204Search in Google Scholar
[14] Ellis, R.—Gottschalk, W. H.: Homomorphisms of transformation groups, Trans. Amer. Math. Soc. 94 (1960), 258–271.10.1090/S0002-9947-1960-0123635-1Search in Google Scholar
[15] Florescu, L. C.: Probabilistic convergence structures, Aequat. Math. 38 (1989), 123–145.10.1007/BF01839999Search in Google Scholar
[16] Ford, L. R.: Homeomorphism groups and coset spaces, Trans. Amer. Math. Soc. 77 (1954), 490–497.10.1090/S0002-9947-1954-0066636-1Search in Google Scholar
[17] Fritsche, R.: Topologies for probabilistic metric spaces, Fund. Math. 72 (1971), 7–16.10.4064/fm-72-1-7-16Search 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.2014.04.013Search 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.2010.10.026Search 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.Search 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-XSearch 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/S0218488509005887Search in Google Scholar
[23] Jäger, G.—Ahsanullah, T. M. G.: Probabilistic limit groups under a t-norm, Topology Proc. 44 (2014), 59–74.Search in Google Scholar
[24] Jäger, G.: A convergence theory for probabilistic metric spaces, Quaest. Math. 38 (2015), 587–599.10.2989/16073606.2014.981734Search in Google Scholar
[25] Lipovan, O.: Summeasures with probabilistic structures, Math. Morav. 4 (2000), 59–65.10.5937/MatMor0004059LSearch in Google Scholar
[26] Menger, K.: Statistical metrics, Proc. Nat. Acad. Sci. USA 28 (1942), 535–537.10.1007/978-3-7091-6045-9_35Search in Google Scholar
[27] Montgomery, D.—Zippin, L.: Topological Transformation Groups, Interscience Publishers, Inc., New York, 1955.Search in Google Scholar
[28] Nusser, H.: A generalization of probabilistic uniform spaces, Appl. Categor. Struct. 10 (2002), 81–98.10.1023/A:1013375301613Search in Google Scholar
[29] Park, W. R.: Convergence structures on homeomorphism groups, Math. Ann. 199 (1972), 45–54.10.1007/BF01419575Search in Google Scholar
[30] Pfanzagl, J.: Transformation groups and sufficient statistics, Ann. Math. Stat. 43 (1972), 553–568.10.1214/aoms/1177692636Search in Google Scholar
[31] Poppe, H.: Compactness in General Function Spaces, VEB Deutscher Verlag der Wissenschaften, Berlin, 1974.Search in Google Scholar
[32] Preuss, G.: Foundations of Topology: An Approach to Convenient Topology, Kluwer Academic Publishers, Dordrecht, 2002.10.1007/978-94-010-0489-3Search in Google Scholar
[33] Rath, N.: Action of convergence groups, Topology Proc. 27(2) (2003), 601–1612.Search in Google Scholar
[34] Richardson, G. D.—Kent, D. C.: Probabilistic convergence spaces, J. Austral. Math. Soc. 61 (1996), 400–420.10.1017/S1446788700000483Search in Google Scholar
[35] Richardson, G. D.: Convergence in probabilistic semimetric spaces, Rocky Mountain J. Math. 18(3) (1988), 617–634.10.1216/RMJ-1988-18-3-617Search in Google Scholar
[36] Saminger, S.—Sempi, C.: A primer on triangle functions I, Aequat. Math. 76 (2008), 201–240.10.1007/s00010-008-2936-8Search in Google Scholar
[37] Schweizer, B.—Sklar, A.: Probabilistic Metric Spaces, North-Holland, New York, 1983.Search 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-xSearch 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.Search 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/s1-44.1.441Search 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-1971-1-3-427Search in Google Scholar
[42] Tardiff, R. M.: Topologies for probabilistic metric spaces, Pacific J. Math. 65 (1976), 233–251.10.2140/pjm.1976.65.233Search in Google Scholar
[43] Thorp, E.: Generalized topologies for statistical metric spaces, Fund. Math. 51 (1962), 9–12.10.4064/fm-51-1-9-21Search in Google Scholar
[44] De Vries, J.: Topological Transformtion Groups 1. A Categorical Approach. Mathematical Centre Tracts 65, Mathematisch Centrum, Amsterdam, 1975.Search 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.29.6.196Search in Google Scholar PubMed PubMed Central
[46] Yang, J. S.: Transformation groups of automorphisms of 𝒞(X, G), Proc. Amer. Math. Soc. 39 (1973), 619–624.10.2307/2039605Search 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-XSearch in Google Scholar
© 2018 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- 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
Articles in the same Issue
- 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
