The Vitali convergence theorem for the vector-valued McShane integral

Richard Reynolds; Charles W. Swartz

Mathematica Bohemica (2004)

  • Volume: 129, Issue: 2, page 159-176
  • ISSN: 0862-7959

Abstract

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The classical Vitali convergence theorem gives necessary and sufficient conditions for norm convergence in the space of Lebesgue integrable functions. Although there are versions of the Vitali convergence theorem for the vector valued McShane and Pettis integrals given by Fremlin and Mendoza, these results do not involve norm convergence in the respective spaces. There is a version of the Vitali convergence theorem for scalar valued functions defined on compact intervals in ℝ n given by Kurzweil and Schwabik, but again this version does not consider norm convergence in the space of integrable functions. In this paper we give a version of the Vitali convergence theorem for norm convergence in the space of vector-valued McShane integrable functions.

How to cite

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Reynolds, Richard, and Swartz, Charles W.. "The Vitali convergence theorem for the vector-valued McShane integral." Mathematica Bohemica 129.2 (2004): 159-176. <http://eudml.org/doc/249391>.

@article{Reynolds2004,
abstract = {The classical Vitali convergence theorem gives necessary and sufficient conditions for norm convergence in the space of Lebesgue integrable functions. Although there are versions of the Vitali convergence theorem for the vector valued McShane and Pettis integrals given by Fremlin and Mendoza, these results do not involve norm convergence in the respective spaces. There is a version of the Vitali convergence theorem for scalar valued functions defined on compact intervals in $\mathbb \{R\}^\{n\}$ given by Kurzweil and Schwabik, but again this version does not consider norm convergence in the space of integrable functions. In this paper we give a version of the Vitali convergence theorem for norm convergence in the space of vector-valued McShane integrable functions.},
author = {Reynolds, Richard, Swartz, Charles W.},
journal = {Mathematica Bohemica},
keywords = {vector-valued McShane integral; Vitali theorem; norm convergence; vector-valued McShane integral; Vitali convergence theorem; norm convergence},
language = {eng},
number = {2},
pages = {159-176},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Vitali convergence theorem for the vector-valued McShane integral},
url = {http://eudml.org/doc/249391},
volume = {129},
year = {2004},
}

TY - JOUR
AU - Reynolds, Richard
AU - Swartz, Charles W.
TI - The Vitali convergence theorem for the vector-valued McShane integral
JO - Mathematica Bohemica
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 129
IS - 2
SP - 159
EP - 176
AB - The classical Vitali convergence theorem gives necessary and sufficient conditions for norm convergence in the space of Lebesgue integrable functions. Although there are versions of the Vitali convergence theorem for the vector valued McShane and Pettis integrals given by Fremlin and Mendoza, these results do not involve norm convergence in the respective spaces. There is a version of the Vitali convergence theorem for scalar valued functions defined on compact intervals in $\mathbb {R}^{n}$ given by Kurzweil and Schwabik, but again this version does not consider norm convergence in the space of integrable functions. In this paper we give a version of the Vitali convergence theorem for norm convergence in the space of vector-valued McShane integrable functions.
LA - eng
KW - vector-valued McShane integral; Vitali theorem; norm convergence; vector-valued McShane integral; Vitali convergence theorem; norm convergence
UR - http://eudml.org/doc/249391
ER -

References

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  1. Linear Operators, Interscience, N.Y., 1958. (1958) 
  2. 10.1215/ijm/1255986891, Illinois J. Math. 38 (1994), 127–147. (1994) MR1245838DOI10.1215/ijm/1255986891
  3. 10.1215/ijm/1255986628, Illinois J. Math. 39 (1995), 39–67. (1995) Zbl0810.28006MR1299648DOI10.1215/ijm/1255986628
  4. 10.1215/ijm/1255988170, Illinois J. Math. 34 (1990), 557–567. (1990) Zbl0685.28003MR1053562DOI10.1215/ijm/1255988170
  5. Some comments on the McShane and Henstock integrals, Real Anal. Exchange 23 (1997/98), 329–341. (1997/98) MR1609917
  6. Real and Abstract Analysis, Springer, N.Y., 1965. (1965) MR0367121
  7. On the McShane integrability of Banach space-valued functions, (to appear). (to appear) MR2083811
  8. McShane equi-integrability and Vitali’s convergence theorem, Math. Bohem. 129 (2004), 141–157. (2004) MR2073511
  9. Unified Integration, Academic Press, N.Y., 1983. (1983) Zbl0551.28001MR0740710
  10. Topics in the theory of Pettis integration, Rendiconti Inst. Mat. Univ. Trieste 23 (1991), 177–262. (1991) Zbl0798.46042MR1248654
  11. The Generalized McShane Integral for Vector-Valued Functions, Ph.D. dissertation, New Mexico State University, 1997. (1997) 
  12. Real Analysis, Macmillan, N.Y., 1988. (1988) Zbl0704.26006MR0151555
  13. Measure, Integration, and Function Spaces, World Scientific, Singapore, 1994. (1994) Zbl0814.28001MR1337502
  14. 10.36045/bbms/1105737762, Bull. Belgian Math. Soc. 4 (1997), 589–599. (1997) Zbl1038.46505MR1600292DOI10.36045/bbms/1105737762
  15. Uniform integrability and mean convergence for the vector-valued McShane integral, Real Anal. Exchange 23 (1997/98), 303–312. (1997/98) MR1609766
  16. Introduction to Gauge Integrals, World Scientific, Singapore, 2001. (2001) Zbl0982.26006MR1845270

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