| Title:
|
Convergence theorems for the Birkhoff integral (English) |
| Author:
|
Balcerzak, Marek |
| Author:
|
Potyrała, Monika |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
4 |
| Year:
|
2008 |
| Pages:
|
1207-1219 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $f_n\to f$ almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of $(f_n)$ to $f$ is assumed. (English) |
| Keyword:
|
Birkhoff integral |
| Keyword:
|
convergence theorems |
| Keyword:
|
vector valued functions |
| MSC:
|
28B05 |
| MSC:
|
46G10 |
| idZBL:
|
Zbl 1174.28011 |
| idMR:
|
MR2471177 |
| . |
| Date available:
|
2010-07-21T08:15:56Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140451 |
| . |
| Reference:
|
[1] Birkhoff, G.: Integration of functions with values in a Banach space.Trans. Amer. Math. Soc. 38 (1935), 357-378. Zbl 0013.00803, MR 1501815 |
| Reference:
|
[2] Cascales, B., Rodríguez, J.: The Birkhoff integral and the property of Bourgain.Math. Ann. 331 (2005), 259-279. MR 2115456, 10.1007/s00208-004-0581-7 |
| Reference:
|
[3] J. Diestel, J. J. Uhl., Jr.: Vector measures.Math. Surveys, 15, Amer. Math. Soc., Providence, Rhode Island (1977). Zbl 0369.46039, MR 0453964 |
| Reference:
|
[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. |
| Reference:
|
[5] Hille, E., Phillips, R.: Functional Analysis and Semi-Groups.Colloquium Publications, 31, Amer. Math. Soc., Providence, Rhode Island (1957). MR 0089373 |
| Reference:
|
[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. Zbl 1084.28008, MR 1911073 |
| Reference:
|
[7] Kadets, V. M., Tseytlin, L. M.: On `integration' of non-integrable vector-valued functions.Mat. Fiz. Anal. Geom. 7 (2000), 49-65. Zbl 0974.28007, MR 1760946 |
| Reference:
|
[8] Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I, Sequence Spaces.Springer-Verlag, Berlin, Heidelberg, New York (1977). Zbl 0362.46013, MR 0500056 |
| Reference:
|
[9] Marraffa, V.: A characterization of absolutely summing operators by means of McShane integrable functions.J. Math. Anal. Appl. 293 (2004), 71-78. Zbl 1087.47023, MR 2052532, 10.1016/j.jmaa.2003.12.029 |
| Reference:
|
[10] Potyrała, M.: Some remarks about Birkhoff and Riemann-Lebesgue integrability of vector valued functions.Tatra Mt. Math. Publ. 35 (2007), 97-106. MR 2372438 |
| Reference:
|
[11] Potyrała, M.: The Birkhoff and variational McShane integrals of vector valued functions.Folia Mathematica, Acta Universitatis Lodziensis 13 (2006), 31-40. MR 2675441 |
| Reference:
|
[12] Rodríguez, J.: 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:
|
[13] Rodríguez, J.: On integration of vector functions with respect to vector measures.Czech. Math. J. 56 (2006), 805-825. MR 2261655, 10.1007/s10587-006-0058-9 |
| Reference:
|
[14] Schwabik, Š., Guoju, Ye: Topics in Banach Space Integration.Series in Real Analysis, 10, World Scientific, Singapore (2005). MR 2167754 |
| . |
