| Title:
|
The McShane, PU and Henstock integrals of Banach valued functions (English) |
| Author:
|
Di Piazza, Luisa |
| Author:
|
Marraffa, V. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
52 |
| Issue:
|
3 |
| Year:
|
2002 |
| Pages:
|
609-633 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational integrals of vector valued functions are characterized. (English) |
| Keyword:
|
Pettis |
| Keyword:
|
McShane |
| Keyword:
|
PU and Henstock integrals |
| Keyword:
|
variational integrals |
| Keyword:
|
multipliers |
| MSC:
|
26A39 |
| MSC:
|
26B30 |
| MSC:
|
28B05 |
| MSC:
|
46G10 |
| idZBL:
|
Zbl 1011.28007 |
| idMR:
|
MR1923266 |
| . |
| Date available:
|
2009-09-24T10:54:49Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127748 |
| . |
| Reference:
|
[1] B. Bongiorno: Relatively weakly compact sets in the Denjoy space.J. Math. Study 27 (1994), 37–43. Zbl 1045.26502, MR 1318256 |
| Reference:
|
[2] B. Bongiorno and L. Di Piazza: Convergence theorem for generalized Riemann-Stieltjes integrals.Real Anal. Exchange 17 (1991–92), 339–361. MR 1147373 |
| Reference:
|
[3] B. Bongiorno, M. Giertz and W. Pfeffer: Some nonabsolutely convergent integrals in the real line.Boll. Un. Mat. Ital. (7) 6-B (1992), 371–402. MR 1171108 |
| Reference:
|
[4] B. Bongiorno and W. Pfeffer: A concept of absolute continuity and a Riemann type integral.Comment. Math. Univ. Carolin. 33 (1992), 184–196. MR 1189651 |
| Reference:
|
[5] J. K. Brooks: Representation of weak and strong integrals in Banach spaces.Proc. Nat. Acad. Sci., U.S.A. (1969), 266–279. MR 0274697 |
| Reference:
|
[6] S. Cao: The Henstock integral for Banach-valued functions.SEA Bull. Math. 16 (1992), 35–40. Zbl 0749.28007, MR 1173605 |
| Reference:
|
[7] D. Caponetti and V. Marraffa: An integral in the real line defined by BV partitions of unity.Atti Sem. Mat. Fis. Univ. Modena XlII (1994), 69–82. MR 1282323 |
| Reference:
|
[8] J. Diestel and J. J. Uhl Jr.: Vector Mesures. Mathematical Surveys, No.15.Amer. Math. Soc., 1977. MR 0453964 |
| Reference:
|
[9] W. Congxin and Y. Xiaobo: A Riemann-type definition of the Bochner integral.J. Math. Study 27 (1994), 32–36. MR 1318255 |
| Reference:
|
[10] D. H. Fremlin: On the Henstock and McShane integrals of vector-valued functions.Illinois J. Math. 38 (1994), 471–479. MR 1269699, 10.1215/ijm/1255986726 |
| Reference:
|
[11] D. H. Fremlin and J. Mendoza: On the integration of vector-valued functions.Illinois J. Math. 38 (1994), 127–147. MR 1245838, 10.1215/ijm/1255986891 |
| Reference:
|
[12] R. Gordon: Riemann integration in Banach spaces.Rocky Mountain J. Math. 21 (1991), 923–949. Zbl 0764.28008, MR 1138145, 10.1216/rmjm/1181072923 |
| Reference:
|
[13] E. Hille and R. S. Phillips: Functional Analysis and Semigroups.AMS Colloquium Publications, Vol. XXXI, 1957. |
| Reference:
|
[14] R. C. James: Weak compactness and reflexivity.Israel J. Math. 2 (1964), 101–119. Zbl 0127.32502, MR 0176310, 10.1007/BF02759950 |
| Reference:
|
[15] J. Kurzweil, J. Mawhin and W. F. Pfeffer: An integral defined by approximating BV partitions of unity.Czechoslovak Math. J. 41(116) (1991), 695–712. MR 1134958 |
| Reference:
|
[16] P. Y. Lee: Lanzhou Lectures on Henstock Integration.World Scientific, Singapore, 1989. Zbl 0699.26004, MR 1050957 |
| Reference:
|
[17] V. Marraffa: A descriptive characterization of the variational Henstock integral.Matimyás Mat. 22 (1999), 73–84. Zbl 1030.28005, MR 1770168 |
| Reference:
|
[18] K. Musial: Pettis integration.Suppl. Rend. Circ. Mat. Palermo, Ser. II, 10 (1985), 133–142. Zbl 0649.46040, MR 0894278 |
| Reference:
|
[19] V. A. Skvortsov and A. P. Solodov: A variational integral for Banach-valued functions.Real Anal. Exchange 24 (1998–99), 799–806. MR 1704751 |
| Reference:
|
[20] B. S. Thomson: Derivatives of Interval Functions.Memoires of the American Mathematical Society No. 452, 1991. MR 1078198 |
| . |
