Two valued measure and summability of double sequences

  • Pratulananda Das
  • Santanu Bhunia


In this paper, following the methods of Connor [2], we extend the idea of statistical convergence of a double sequence (studied by Muresaleen and Edely [12]) to μ-statistical convergence and convergence in μ-density using a two valued measure μ. We also apply the same methods to extend the ideas of divergence and Cauchy criteria for double sequences. We then introduce a property of the measure μ called the (APO2) condition, inspired by the (APO) condition of Connor [3]. We mainly investigate the interrelationships between the two types of convergence, divergence and Cauchy criteria and ultimately show that they become equivalent if and only if the measure μ has the condition (APO2).


double sequences μ-statistical convergence divergence and Cauchy criteria convergence divergence and Cauchy criteria in μ-density condition (APO2

MSC 2000

40A30 40A05 


  1. [1]
    M. Balcerzak and K. Dems: Some types of convergence and related Baire systems. Real Anal. Exchange 30 (2004), 267–276.MathSciNetGoogle Scholar
  2. [2]
    J. Connor: Two valued measure and summability. Analysis 10 (1990), 373–385.MATHMathSciNetGoogle Scholar
  3. [3]
    J. Connor: R-type summability methods, Cauchy criterion, P-sets and statistical convergence. Proc. Amer. Math. Soc. 115 (1992), 319–327.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J. Connor, J. A. Fridy and C. Orhan: Core equality results for sequences. J. Math. Anal. Appl. 321 (2006), 515–523.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    P. Das and P. Malik: On the statistical and I variation of double sequences. Real Anal. Exchange 33 (2008), 351–364.MATHMathSciNetGoogle Scholar
  6. [6]
    P. Das, P. Kostyrko, W. Wilczyński and P. Malik: I and I*-convergence of double sequences. Math. Slovaca 58 (2008), 605–620.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    K. Dems: On I-Cauchy sequences. Real Anal. Exchange 30 (2004), 123–128.MathSciNetGoogle Scholar
  8. [8]
    H. Fast: Sur la convergence statistique. Colloq. Math. 2 (1951), 241–244.MATHMathSciNetGoogle Scholar
  9. [9]
    J. A. Fridy: On statistical convergence. Analysis 5 (1985), 301–313.MATHMathSciNetGoogle Scholar
  10. [10]
    P. Kostyrko, T. Šalát and W. Wilczyński: I-Convergence. Real Anal. Exchange 26(2000/2001), 669–686.MathSciNetGoogle Scholar
  11. [11]
    F. Móricz: Statistical convergence of multiple sequences. Arch. Math. 81 (2003), 82–89.MATHCrossRefGoogle Scholar
  12. [12]
    Muresaleen and Osama H.H. Edely: Statistical convergence of double sequences. J. Math. Anal. Appl. 288 (2003), 223–231.CrossRefMathSciNetGoogle Scholar
  13. [13]
    F. Nuray and W. H. Ruckle: Generalized statistical convergence and convergence free spaces. J. Math. Anal. Appl. 245 (2000), 513–527.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    A. Pringsheim: Zur Theorie der zweifach unendlichen Zahlenfolgen. Math. Ann. 53(1900), 289–321.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    E. Savas, Muresaleen: On statistically convergent double sequences of fuzzy numbers. Information Sciences 162 (2004), 183–192.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    T. Šalát: On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), 139–150.MATHMathSciNetGoogle Scholar
  17. [17]
    I. J. Schoenberg: The integrability of certain functions and related summability methods. Amer. Math. Monthly. 66 (1959), 361–375.CrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2009

Authors and Affiliations

  1. 1.Department of MathematicsJadavpur UniversityKolkataIndia
  2. 2.Department of MathematicsF.C.College, Diamond HarbourPinWest BengalIndia

Personalised recommendations