Functiones et Approximatio Commentarii Mathematici

Nets and sequences of Riemann and Riemann-type integrable functions with values in a Banach space

Sk. Jaker Ali, Lakshmi Kanta Dey, and Pratikshan Mondal

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Abstract

In this article, we discuss several aspects of convergence theorems for nets and sequences of Riemann and Riemann-type integrable functions defined on a closed bounded interval in $\mathbb{R}$ with values in a Banach space. We introduce the notions of Riemann $\Delta$-Cauchy nets of functions with its analogous variants and derive some correlations between such kind of nets of functions and equi-Riemann integrability. Moreover, we establish equi-integrability of the pointwise closure of different types of equi-integrable collections of functions. Finally, several related results, e.g., relative compactness of equi-integrable collections of functions with respect to different topologies are studied.

Article information

Source
Funct. Approx. Comment. Math., Advance publication (2019), 24 pages.

Dates
First available in Project Euclid: 9 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.facm/1573268429

Digital Object Identifier
doi:10.7169/facm/1789

Subjects
Primary: 40A30: Convergence and divergence of series and sequences of functions 28B05: Vector-valued set functions, measures and integrals [See also 46G10]
Secondary: 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22]

Keywords
Riemann $\Delta$-Cauchy equi-integrability exhaustiveness eventually uniformly bounded $p$-Bochner norm $p$-Pettis norm

Citation

Mondal, Pratikshan; Dey, Lakshmi Kanta; Ali, Sk. Jaker. Nets and sequences of Riemann and Riemann-type integrable functions with values in a Banach space. Funct. Approx. Comment. Math., advance publication, 9 November 2019. doi:10.7169/facm/1789. https://projecteuclid.org/euclid.facm/1573268429


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References

  • [1] J. Alan Alewine and E. Schechter, Topologizing the Denjoy space by measuring equiintegrability, Real Anal. Exchange 31(1) (2005/2006), 23–44.
  • [2] M. Balcerzak and K. Musiał, Vitali type convergence theorems for Banach space valued integrals, Acta Math. Sin. (Engl. Ser.) 29(11) (2013), 2027–2036.
  • [3] M. Balcerzak and K. Musiał, A convergence theorem for the Birkhoff integral, Funct. Approx. Comment. Math. 50(1) (2014), 161–168.
  • [4] M. Balcerzak and M. Potyrała, Convergence theorems for the Birkhoff integral, Czechoslovak Math. J. 58(133) (2008), 1207–1219.
  • [5] R.G. Bartle, A convergence theorem for generalized Riemann integrals, Real Anal. Exchange 20(1) (1994/95), 119–124.
  • [6] R.G. Bartle, A Modern Theory of Integration, Graduate Studies in Mathematics, Volume 32, American Mathematical Society, Providence, R.I., 2001.
  • [7] R.R. de Cenzano, An application of Kadets renorming theorem to vector-valued Riemann integrability, RACSAM 108 (2014), 861–865.
  • [8] L. Di Piazza, Kurzweil-Henstock type integration on Banach spaces, Real Anal. Exchange 29(2) (2003/2004), 543–555.
  • [9] L. Di Piazza and K. Musiał, Characterizations of Kurzweil-Henstock-Pettis integrable functions, Studia Math. 176(2) (2006), 159–176.
  • [10] J. Diestel and J.J. Uhl, Jr., Vector Measures, Mathematical Surveys Volume 15, American Mathematical Society, Providence, R.I., 1977.
  • [11] R.A. Gordon, Another look at a convergence theorem for the Henstock integral, Real Anal. Exchange 15 (1989-90), 724–728.
  • [12] R.A. Gordon, Riemann integration in Banach spaces, Rocky Mountain J. Math. 21(3) (1991), 923–949.
  • [13] R.A. Gordon, An iterated limits theorem applied to the Henstock integral, Real Anal. Exchange 21(2) (1995-96), 774–781.
  • [14] R.A. Gordon, A convergence theorem for the Riemann integral, Math. Mag. 73(2) (2000), 141–147.
  • [15] V. Gregoriades and N. Papanastassiou, The notion of exhaustiveness and Ascoli-type theorems, Topology Appl. 155 (2008), 1111–1128.
  • [16] S.F.Y. Lee, Interchange of limit operations and partitions of unity, Academic Exercise (B.Sc. Hons.) National Institute of Education, Nanyang Technological University, 1998. http://hdl.handle.net/10497/2353.
  • [17] P. Mondal, L.K. Dey and Sk.J. Ali, Equi-Riemann and equi-Riemann-type integrable functions with values in a Banach space, Real Anal. Exchange 43(2) (2018), 301–324.
  • [18] P. Mondal, L.K. Dey and Sk.J. Ali, Nets and sequences of $p$-Bochner, $p$-Dunford and $p$-Pettis integrable functions with values in a Banach space - (communicated).
  • [19] K. Musiał, Pettis integration, Proc. 13th Winter School on Abstract Analysis (Srni, 1985), Rend. Circ. Mat. Palermo, Ser II Suppl. 10 (1985), 133–142.
  • [20] J. Rodriguez, Pointwise limits of Birkhoff integrable functions, Proc. Amer. Math. Soc. 137(1) (2009), 235–245.
  • [21] J. Rodriguez, Convergence theorems for the Birkhoff integral, Houston J. Math. 35(2) (2009), 541–551.
  • [22] S. Schwabik and G. Ye, Topics in Banach Space Integration, Series in Real Analysis, Volume 10, World Scientific Publishing Co. Pte. Ltd., 2005.
  • [23] Sk.J. Ali and P. Mondal, Riemann and Riemann-type integration in Banach spaces, Real Anal. Exchange 39(2) (2013/14), 403–440.
  • [24] C. Swartz, Uniform integrability and mean convergence for the vector-valued McShane integral, Real Anal. Exchange 23(1) (1998-99), 303–311.
  • [25] C. Swartz, Norm convergence and uniform integrability for the Henstock-Kurzweil integral, Real Anal. Exchange 24(1) (1998/99), 423–426.
  • [26] C. Wang, On the weak property of Lebesgue of Banach spaces, Journal of Nanjing University (Mathematical Biquarterly) 13(2) (1996), 150–155.