Functiones et Approximatio Commentarii Mathematici

A convergence theorem for the Birkhoff integral

Marek Balcerzak and Kazimierz Musiał

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We propose an essential improvement of a convergence theorem for the Birkhoff integral. We also obtain the respective version of this result for the convergence associated with an ideal on $\mathbb N$.

Article information

Funct. Approx. Comment. Math., Volume 50, Number 1 (2014), 161-168.

First available in Project Euclid: 27 March 2014

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Zentralblatt MATH identifier

Primary: 11K06: General theory of distribution modulo 1 [See also 11J71]
Secondary: 11K38: Irregularities of distribution, discrepancy [See also 11Nxx] 11K41: Continuous, $p$-adic and abstract analogues

convergence theorems for integrals Pettis integral Birkhoff integral


Balcerzak, Marek; Musiał, Kazimierz. A convergence theorem for the Birkhoff integral. Funct. Approx. Comment. Math. 50 (2014), no. 1, 161--168. doi:10.7169/facm/2014.50.1.5.

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  • M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007), 715–729.
  • M. Balcerzak, K. Musiał, Vitali type convergence theorems for Banach space valued functions, Acta Math. Sinica, English Ser. 29 (2013), 2027–2036.
  • M. Balcerzak, M. Potyrała, Convergence theorems for the Birkhoff integral, Czechoslovak Math. J. 58 (2008), 1207–1219.
  • G. Birkhoff, Integration of functions with values in a Banach space, Trans. Amer. Math. Soc. 38 (1935), 357–378.
  • B. Cascales, J. Rodríguez, The Birkhoff integral and the property of Bourgain, Math. Ann. 331 (2005), 259–279.
  • J. Diestel, Jr. J.J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, Rhode Island, 1977.
  • G. Di Maio, Lj.D.R. Kočiniac, Statistical convergence in topology, Topology Appl. 156 (2008), 28–45.
  • R. Filipów, N. Mrożek, I. Recław, P. Szuca, Ideal convergence of bounded sequences, J. Symb. Logic 72 (2007), 501–512.
  • V. Kadets, A. Leonov, Dominated convergence and Egorov theorems for filter convergence, J. Math. Phys. Anal. Geom. 3(2) (2007), 196–212.
  • V. Kadets, A. Leonov, Weak and point-wise convergence in $C(K)$ for filter convergence, J. Math. Anal. Appl. 350 (2009), 455–463.
  • V. Kadets, A. Leonov, C. Orhan, Weak statistical convergence and weak filter convergence for unbounded sequences, J. Math. Anal. Appl. 371 (2010), 414–424.
  • A. Komisarski, Pointwise $\mathcal I$-convergence and $\mathcal I$-convergence in measure of sequences of functions, J. Math. Anal. Appl. 340 (2008), 770–779.
  • P. Kostyrko, T. Šalát, W. Wilczyński, $\mathcal I$-Convergence, Real Anal. Exchange 26 (2000/2001), 669–689.
  • K. Musiał, Functions with values in a Banach space possessing the Radon-Nikodym property, Aarhus University, Preprint Series 29 (1977).
  • K. Musiał, Martingales of Pettis Integrable Functions, Proc. Conf. Measure Theory (Oberwolfach 1979), Lecture Notes in Math. 794 (1980), 324–339.
  • K. Musiał, Pettis integration, Suppl. Rend. Circolo Mat. di Palermo, Ser II, 10 (1985), 133–142.
  • K. Musiał, Topics in the theory of Pettis integration, Rend. Istit. Mat. Univ. Trieste (School on Measure Theory and Real Analysis, Grado, 1991) 23 (1991), 177–262.
  • K. Musiał, Pettis integral, in: E. Pap (Ed.), Handbook of Measure Theory, Vol. I, II, North-Holland, Amsterdam, 2002, 531–586.
  • K. Musiał, Pettis integrability of multifunctions with values in arbitrary Banach spaces, J. Convex Anal. 18 (2011), 769–810.
  • F. Nuray, W.H. Ruckle, Generalized statistical convergence and convergence free spaces, J. Math. Anal. Appl. 245 (2000), 513–527.
  • L.H. Riddle, E. Saab, On functions that are universally Pettis integrable, Illinois J. Math. 29 (1985), 509–531.
  • J. Rodríguez, On the existence of Pettis integrable functions which are not Birkhoff integrable, Proc. Amer. Math. Soc. 133 (2005), 1157–1163.
  • J. Rodríguez, Convergence theorems for the Birkhoff integral, Houston J. Math. 35 (2009), 541–551.
  • J. Rodríguez, Pointwise limits of Birkhoff integrable functions, Proc. Amer. Math. Soc. 137 (2009), 235–245.