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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  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
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