| Title:
|
On integration of vector functions with respect to vector measures (English) |
| Author:
|
Rodríguez, José |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
56 |
| Issue:
|
3 |
| Year:
|
2006 |
| Pages:
|
805-825 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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). (English) |
| Keyword:
|
Bartle $^*$-integral |
| Keyword:
|
Dobrakov integral |
| Keyword:
|
McShane integral |
| Keyword:
|
Birkhoff integral |
| Keyword:
|
$S^*$-integral |
| MSC:
|
28B05 |
| MSC:
|
46G10 |
| idZBL:
|
Zbl 1164.28305 |
| idMR:
|
MR2261655 |
| . |
| Date available:
|
2009-09-24T11:38:23Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128108 |
| . |
| Reference:
|
[1] R. G. Bartle: A general bilinear vector integral.Studia Math. 15 (1956), 337–352. Zbl 0070.28102, MR 0080721, 10.4064/sm-15-3-337-352 |
| Reference:
|
[2] G. Birkhoff: Integration of functions with values in a Banach space.Trans. Amer. Math. Soc. 38 (1935), 357–378. Zbl 0013.00803, MR 1501815 |
| Reference:
|
[3] B. Cascales, J. Rodríguez: The Birkhoff integral and the property of Bourgain.Math. Ann. 331 (2005), 259–279. MR 2115456, 10.1007/s00208-004-0581-7 |
| Reference:
|
[4] L. Di Piazza, D. Preiss: When do McShane and Pettis integrals coincide? Illinois J.Math. 47 (2003), 1177–1187. MR 2036997, 10.1215/ijm/1258138098 |
| Reference:
|
[5] J. Diestel, J. J. Uhl, Jr.: Vector Measures. Mathematical Surveys, No. 15.American Mathematical Society, Providence, 1977. MR 0453964 |
| Reference:
|
[6] I. Dobrakov: On integration in Banach spaces I.Czechoslovak Math. J. 20(95) (1970), 511–536. Zbl 0215.20103, MR 0365138 |
| Reference:
|
[7] I. Dobrakov: On representation of linear operators on $C_0(T,{\mathrm X})$.Czechoslovak Math. J. 21(96) (1971), 13–30. MR 0276804 |
| Reference:
|
[8] I. Dobrakov: On integration in Banach spaces VII.Czechoslovak Math. J. 38(113) (1988), 434–449. Zbl 0674.28003, MR 0950297 |
| Reference:
|
[9] I. Dobrakov, P. Morales: On integration in Banach spaces VI.Czechoslovak Math. J. 35(110) (1985), 173–187. MR 0787123 |
| Reference:
|
[10] N. Dunford, J. T. Schwartz: 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. MR 1009162 |
| Reference:
|
[11] D. H. Fremlin: Four problems in measure theory. Version of 30.7.03.Available at URL http://www.essex.ac.uk/maths/staff/fremlin/preprints.htm. MR 1258546 |
| Reference:
|
[12] D. H. Fremlin: 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. |
| Reference:
|
[13] D. H. Fremlin: Problem ET, version of 27.10.04.Available at URL http://www.essex.ac.uk/maths/staff/fremlin/problems.htm. |
| Reference:
|
[14] D. H. Fremlin: The generalized McShane integral.Illinois J. Math. 39 (1995), 39–67. Zbl 0810.28006, MR 1299648, 10.1215/ijm/1255986628 |
| Reference:
|
[15] D. H. Fremlin: Measure Theory, Vol. 4: Topological Measure Spaces.Torres Fremlin, Colchester, 2003. MR 2462372 |
| Reference:
|
[16] D. H. Fremlin, J. Mendoza: On the integration of vector-valued functions.Illinois J. Math. 38 (1994), 127–147. MR 1245838, 10.1215/ijm/1255986891 |
| Reference:
|
[17] F. J. Freniche, J. C. García-Vázquez: The Bartle bilinear integration and Carleman operators.J. Math. Anal. Appl. 240 (1999), 324–339. MR 1731648, 10.1006/jmaa.1999.6575 |
| Reference:
|
[18] T. H. Hildebrandt: Integration in abstract spaces.Bull. Amer. Math. Soc. 59 (1953), 111–139. Zbl 0051.04201, MR 0053191, 10.1090/S0002-9904-1953-09694-X |
| Reference:
|
[19] B. Jefferies, S. Okada: Bilinear integration in tensor products.Rocky Mountain J. Math. 28 (1998), 517–545. MR 1651584, 10.1216/rmjm/1181071785 |
| Reference:
|
[20] A. N. Kolmogorov: Untersuchungen über Integralbegriff.Math. Ann. 103 (1930), 654–696. 10.1007/BF01455714 |
| Reference:
|
[21] R. Pallu de La Barrière: Integration of vector functions with respect to vector measures.Studia Univ. Babeş-Bolyai Math. 43 (1998), 55–93. MR 1855339 |
| Reference:
|
[22] T. V. Panchapagesan: On the distinguishing features of the Dobrakov integral.Divulg. Mat. 3 (1995), 79–114. Zbl 0883.28011, MR 1374668 |
| Reference:
|
[23] J. Rodríguez: On the existence of Pettis integrable functions which are not Birkhoff integrable.Proc. Amer. Math. Soc. 133 (2005), 1157–1163. MR 2117218, 10.1090/S0002-9939-04-07665-8 |
| Reference:
|
[24] G. F. Stefánsson: Integration in vector spaces.Illinois J. Math. 45 (2001), 925–938. MR 1879244, 10.1215/ijm/1258138160 |
| Reference:
|
[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. MR 1175399 |
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