On integration of vector functions with respect to vector measures

José Rodríguez

Czechoslovak Mathematical Journal (2006)

  • Volume: 56, Issue: 3, page 805-825
  • ISSN: 0011-4642

Abstract

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We study integration of Banach space-valued functions with respect to Banach space-valued measures. We focus our attention on natural extensions to this setting of the Birkhoff and McShane integrals. The corresponding generalization of the Birkhoff integral was first considered by Dobrakov under the name S * -integral. Our main result states that S * -integrability implies McShane integrability in contexts in which the later notion is definable. We also show that a function is measurable and McShane integrable if and only if it is Dobrakov integrable (i.e. Bartle *-integrable).

How to cite

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Rodríguez, José. "On integration of vector functions with respect to vector measures." Czechoslovak Mathematical Journal 56.3 (2006): 805-825. <http://eudml.org/doc/31069>.

@article{Rodríguez2006,
abstract = {We study integration of Banach space-valued functions with respect to Banach space-valued measures. We focus our attention on natural extensions to this setting of the Birkhoff and McShane integrals. The corresponding generalization of the Birkhoff integral was first considered by Dobrakov under the name $S^\{*\}$-integral. Our main result states that $S^\{*\}$-integrability implies McShane integrability in contexts in which the later notion is definable. We also show that a function is measurable and McShane integrable if and only if it is Dobrakov integrable (i.e. Bartle *-integrable).},
author = {Rodríguez, José},
journal = {Czechoslovak Mathematical Journal},
keywords = {Bartle $^*$-integral; Dobrakov integral; McShane integral; Birkhoff integral; $S^*$-integral; Bartle -integral; Dobrakov integral; McShane integral; Birkhoff integral; -integral},
language = {eng},
number = {3},
pages = {805-825},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On integration of vector functions with respect to vector measures},
url = {http://eudml.org/doc/31069},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Rodríguez, José
TI - On integration of vector functions with respect to vector measures
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 3
SP - 805
EP - 825
AB - We study integration of Banach space-valued functions with respect to Banach space-valued measures. We focus our attention on natural extensions to this setting of the Birkhoff and McShane integrals. The corresponding generalization of the Birkhoff integral was first considered by Dobrakov under the name $S^{*}$-integral. Our main result states that $S^{*}$-integrability implies McShane integrability in contexts in which the later notion is definable. We also show that a function is measurable and McShane integrable if and only if it is Dobrakov integrable (i.e. Bartle *-integrable).
LA - eng
KW - Bartle $^*$-integral; Dobrakov integral; McShane integral; Birkhoff integral; $S^*$-integral; Bartle -integral; Dobrakov integral; McShane integral; Birkhoff integral; -integral
UR - http://eudml.org/doc/31069
ER -

References

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  1. 10.4064/sm-15-3-337-352, Studia Math. 15 (1956), 337–352. (1956) Zbl0070.28102MR0080721DOI10.4064/sm-15-3-337-352
  2. Integration of functions with values in a Banach space, Trans. Amer. Math. Soc. 38 (1935), 357–378. (1935) Zbl0013.00803MR1501815
  3. 10.1007/s00208-004-0581-7, Math. Ann. 331 (2005), 259–279. (2005) MR2115456DOI10.1007/s00208-004-0581-7
  4. 10.1215/ijm/1258138098, Math. 47 (2003), 1177–1187. (2003) MR2036997DOI10.1215/ijm/1258138098
  5. Vector Measures. Mathematical Surveys, No.  15, American Mathematical Society, Providence, 1977. (1977) MR0453964
  6. On integration in Banach spaces I, Czechoslovak Math. J. 20(95) (1970), 511–536. (1970) Zbl0215.20103MR0365138
  7. On representation of linear operators on  C 0 ( T , X ) , Czechoslovak Math.  J. 21(96) (1971), 13–30. (1971) MR0276804
  8. On integration in Banach spaces VII, Czechoslovak Math.  J. 38(113) (1988), 434–449. (1988) Zbl0674.28003MR0950297
  9. On integration in Banach spaces VI, Czechoslovak Math.  J. 35(110) (1985), 173–187. (1985) MR0787123
  10. Linear Operators. Part I, Wiley Classics Library, John Wiley & Sons, New York, 1988, General theory, with the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958  original, A Wiley-Interscience Publication. (1988) MR1009162
  11. Four problems in measure theory. Version of 30.7.03, Available at URL http://www.essex.ac.uk/maths/staff/fremlin/preprints.htm. MR1258546
  12. The McShane and Birkhoff integrals of vector-valued functions, University of Essex Mathematics Department Research Report 92-10, version of 13.10.04. Available at URL http://www.essex.ac.uk/maths/staff/fremlin/preprints.htm. 
  13. Problem ET, version of 27.10.04, Available at URL http://www.essex.ac.uk/maths/staff/fremlin/problems.htm. 
  14. 10.1215/ijm/1255986628, Illinois J.  Math. 39 (1995), 39–67. (1995) Zbl0810.28006MR1299648DOI10.1215/ijm/1255986628
  15. Measure Theory, Vol.  4: Topological Measure Spaces, Torres Fremlin, Colchester, 2003. (2003) MR2462372
  16. 10.1215/ijm/1255986891, Illinois J.  Math. 38 (1994), 127–147. (1994) MR1245838DOI10.1215/ijm/1255986891
  17. 10.1006/jmaa.1999.6575, J.  Math. Anal. Appl. 240 (1999), 324–339. (1999) MR1731648DOI10.1006/jmaa.1999.6575
  18. 10.1090/S0002-9904-1953-09694-X, Bull. Amer. Math. Soc. 59 (1953), 111–139. (1953) Zbl0051.04201MR0053191DOI10.1090/S0002-9904-1953-09694-X
  19. 10.1216/rmjm/1181071785, Rocky Mountain J.  Math. 28 (1998), 517–545. (1998) MR1651584DOI10.1216/rmjm/1181071785
  20. 10.1007/BF01455714, Math. Ann. 103 (1930), 654–696. (1930) DOI10.1007/BF01455714
  21. Integration of vector functions with respect to vector measures, Studia Univ. Babeş-Bolyai Math. 43 (1998), 55–93. (1998) MR1855339
  22. On the distinguishing features of the Dobrakov integral, Divulg. Mat. 3 (1995), 79–114. (1995) Zbl0883.28011MR1374668
  23. 10.1090/S0002-9939-04-07665-8, Proc. Amer. Math. Soc. 133 (2005), 1157–1163. (2005) MR2117218DOI10.1090/S0002-9939-04-07665-8
  24. 10.1215/ijm/1258138160, Illinois J.  Math. 45 (2001), 925–938. (2001) MR1879244DOI10.1215/ijm/1258138160
  25. Selected Works of A. N.  Kolmogorov. Vol. I: Mathematics and Mechanics, Mathematics and its Applications (Soviet Series), Vol. 25, V. M.  Tikhomirov (ed.), Kluwer Academic Publishers Group, Dordrecht, 1991, with commentaries by V. I.  Arnol’d, V. A.  Skvortsov, P. L.  Ul’yanov et al. Translated from the Russian original by V. M.  Volosov. (1991) MR1175399

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