Convergence theorems for the Birkhoff integral

Marek Balcerzak; Monika Potyrała

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 4, page 1207-1219
  • ISSN: 0011-4642

Abstract

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We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence ( f n ) of functions from a measure space to a Banach space. In one result the equi-integrability of f n ’s is involved and we assume f n → f almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of ( f n ) to f is assumed.

How to cite

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Balcerzak, Marek, and Potyrała, Monika. "Convergence theorems for the Birkhoff integral." Czechoslovak Mathematical Journal 58.4 (2008): 1207-1219. <http://eudml.org/doc/37897>.

@article{Balcerzak2008,
abstract = {We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence $(f_n)$ of functions from a measure space to a Banach space. In one result the equi-integrability of $f_n$’s is involved and we assume $f_n\rightarrow f$ almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of $(f_n)$ to $f$ is assumed.},
author = {Balcerzak, Marek, Potyrała, Monika},
journal = {Czechoslovak Mathematical Journal},
keywords = {Birkhoff integral; convergence theorems; vector valued functions; Birkhoff integral; convergence theorems; vector valued functions},
language = {eng},
number = {4},
pages = {1207-1219},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Convergence theorems for the Birkhoff integral},
url = {http://eudml.org/doc/37897},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Balcerzak, Marek
AU - Potyrała, Monika
TI - Convergence theorems for the Birkhoff integral
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 4
SP - 1207
EP - 1219
AB - We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence $(f_n)$ of functions from a measure space to a Banach space. In one result the equi-integrability of $f_n$’s is involved and we assume $f_n\rightarrow f$ almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of $(f_n)$ to $f$ is assumed.
LA - eng
KW - Birkhoff integral; convergence theorems; vector valued functions; Birkhoff integral; convergence theorems; vector valued functions
UR - http://eudml.org/doc/37897
ER -

References

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  1. Birkhoff, G., Integration of functions with values in a Banach space, Trans. Amer. Math. Soc. 38 (1935), 357-378. (1935) Zbl0013.00803MR1501815
  2. Cascales, B., Rodríguez, J., 10.1007/s00208-004-0581-7, Math. Ann. 331 (2005), 259-279. (2005) MR2115456DOI10.1007/s00208-004-0581-7
  3. J. Diestel, J. J. Uhl., Jr., Vector measures, Math. Surveys, 15, Amer. Math. Soc., Providence, Rhode Island (1977). (1977) Zbl0369.46039MR0453964
  4. Fremlin, D. H., The McShane and Birkhoff integrals of vector-valued functions, University of Essex, Mathematics Department Reaearch, 1999, Report 92-10, available at http://www.essex.ac.uk/maths/staff/fremlin/preprints.htm. 
  5. Hille, E., Phillips, R., Functional Analysis and Semi-Groups, Colloquium Publications, 31, Amer. Math. Soc., Providence, Rhode Island (1957). (1957) MR0089373
  6. Kadets, V. M., Shumyatskiy, B., Shvidkoy, R., Tseytlin, L., Zheltukhin, K., Some remarks on vector-valued integration, Mat. Fiz. Anal. Geom. 9 (2002), 48-65. (2002) Zbl1084.28008MR1911073
  7. Kadets, V. M., Tseytlin, L. M., On `integration' of non-integrable vector-valued functions, Mat. Fiz. Anal. Geom. 7 (2000), 49-65. (2000) Zbl0974.28007MR1760946
  8. Lindenstrauss, J., Tzafriri, L., Classical Banach Spaces I, Sequence Spaces, Springer-Verlag, Berlin, Heidelberg, New York (1977). (1977) Zbl0362.46013MR0500056
  9. Marraffa, V., 10.1016/j.jmaa.2003.12.029, J. Math. Anal. Appl. 293 (2004), 71-78. (2004) Zbl1087.47023MR2052532DOI10.1016/j.jmaa.2003.12.029
  10. Potyrała, M., Some remarks about Birkhoff and Riemann-Lebesgue integrability of vector valued functions, Tatra Mt. Math. Publ. 35 (2007), 97-106. (2007) MR2372438
  11. Potyrała, M., The Birkhoff and variational McShane integrals of vector valued functions, Folia Mathematica, Acta Universitatis Lodziensis 13 (2006), 31-40. (2006) MR2675441
  12. Rodríguez, J., 10.1090/S0002-9939-04-07665-8, Proc. Amer. Math. Soc. 133 (2005), 1157-1163. (2005) MR2117218DOI10.1090/S0002-9939-04-07665-8
  13. Rodríguez, J., 10.1007/s10587-006-0058-9, Czech. Math. J. 56 (2006), 805-825. (2006) MR2261655DOI10.1007/s10587-006-0058-9
  14. Schwabik, Š., Guoju, Ye, Topics in Banach Space Integration, Series in Real Analysis, 10, World Scientific, Singapore (2005). (2005) MR2167754

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