| Title:
|
Some remarks on descriptive characterizations of the strong McShane integral (English) |
| Author:
|
Kaliaj, Sokol Bush |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
144 |
| Issue:
|
4 |
| Year:
|
2019 |
| Pages:
|
339-355 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We present the full descriptive characterizations of the strong McShane integral (or the variational McShane integral) of a Banach space valued function $f\colon W \to X$ defined on a non-degenerate closed subinterval $W$ of $\mathbb {R}^{m}$ in terms of strong absolute continuity or, equivalently, in terms of McShane variational measure $V_{\mathcal {M}} F$ generated by the primitive $F\colon \mathcal {I}_{W} \to X$ of $f$, where $\mathcal {I}_{W}$ is the family of all closed non-degenerate subintervals of $W$. (English) |
| Keyword:
|
strong McShane integral |
| Keyword:
|
McShane variational measure |
| Keyword:
|
Banach space, $m$-dimensional Euclidean space |
| Keyword:
|
compact non-degenerate $m$-dimensional interval |
| MSC:
|
26A46 |
| MSC:
|
28A35 |
| MSC:
|
28B05 |
| MSC:
|
46B25 |
| MSC:
|
46G10 |
| idZBL:
|
07217259 |
| idMR:
|
MR4047341 |
| DOI:
|
10.21136/MB.2018.0100-17 |
| . |
| Date available:
|
2019-12-09T11:51:19Z |
| Last updated:
|
2020-08-14 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147932 |
| . |
| Reference:
|
[1] Candeloro, D., Piazza, L. Di, Musiał, K., Sambucini, A. R.: Gauge integrals and selections of weakly compact valued multifunctions.J. Math. Anal. Appl. 441 (2016), 293-308. Zbl 1339.28016, MR 3488058, 10.1016/j.jmaa.2016.04.009 |
| Reference:
|
[2] Candeloro, D., Piazza, L. Di, Musiał, K., Sambucini, A. R.: Relations among gauge and Pettis integrals for $cwk(X)$-valued multifunctions.Ann. Mat. Pura Appl. (4) 197 (2018), 171-183. Zbl 06837507, MR 3747527, 10.1007/s10231-017-0674-z |
| Reference:
|
[3] J. Diestel, J. J. Uhl, Jr.: Vector Measures.Mathematical Surveys 15. AMS, Providence (1977). Zbl 0369.46039, MR 0453964 |
| Reference:
|
[4] Piazza, L. Di: Variational measures in the theory of the integration in $\mathbb R^m$.Czech. Math. J. 51 (2001), 95-110. Zbl 1079.28500, MR 1814635, 10.1023/A:1013705821657 |
| Reference:
|
[5] Piazza, L. Di, Musiał, K.: A characterization of variationally McShane integrable Banach-space valued functions.Ill. J. Math. 45 (2001), 279-289. Zbl 0999.28006, MR 1849999, 10.1215/ijm/1258138268 |
| Reference:
|
[6] Dunford, N., Schwartz, J. T.: Linear Operators I. General Theory.Pure and Applied Mathematics. Vol. 7. Interscience Publishers, New York (1958). Zbl 0084.10402, MR 0117523 |
| Reference:
|
[7] Folland, G. B.: Real Analysis. Modern Techniques and Their Applications.Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts. Wiley, New York (1999). Zbl 0924.28001, MR 1681462 |
| Reference:
|
[8] Fremlin, D. H.: The generalized McShane integral.Ill. J. Math. 39 (1995), 39-67. Zbl 0810.28006, MR 1299648, 10.1215/ijm/1255986628 |
| Reference:
|
[9] Gordon, R. A.: The Denjoy extension of the Bochner, Pettis, and Dunford integrals.Stud. Math. 92 (1989), 73-91. Zbl 0681.28006, MR 0984851, 10.4064/sm-92-1-73-91 |
| Reference:
|
[10] Gordon, R. A.: The McShane integral of Banach-valued functions.Ill. J. Math. 34 (1990), 557-567. Zbl 0685.28003, MR 1053562, 10.1215/ijm/1255988170 |
| Reference:
|
[11] Gordon, R. A.: The Integrals of Lebesgue, Denjoy, Perron, and Henstock.Graduate Studies in Mathematics 4. AMS, Providence (1994). Zbl 0807.26004, MR 1288751, 10.1090/gsm/004 |
| Reference:
|
[12] Kaliaj, S. B.: Descriptive characterizations of Pettis and strongly McShane integrals.Real Anal. Exch. 39 (2014), 227-238. Zbl 1298.28025, MR 3261908, 10.14321/realanalexch.39.1.0227 |
| Reference:
|
[13] Lee, T.-Y.: Some full characterizations of the strong McShane integral.Math. Bohem. 129 (2004), 305-312. Zbl 1080.26006, MR 2092716, 10.21136/MB.2004.134144 |
| Reference:
|
[14] Lee, T. Y.: Henstock-Kurzweil Integration on Euclidean Spaces.Series in Real Analysis 12. World Scientific, Hackensack (2011). Zbl 1246.26002, MR 2789724, 10.1142/7933 |
| Reference:
|
[15] Marraffa, V.: The variational McShane integral in locally convex spaces.Rocky Mt. J. Math. 39 (2009), 1993-2013. Zbl 1187.28019, MR 2575890, 10.1216/RMJ-2009-39-6-1993 |
| Reference:
|
[16] McShane, E. J.: Unified Integration.Pure and Applied Mathematics 107. Academic Press, Orlando (Harcourt Brace Jovanovich, Publishers) (1983). Zbl 0551.28001, MR 0740710 |
| Reference:
|
[17] Musiał, K.: Vitali and Lebesgue convergence theorems for Pettis integral in locally convex spaces.Atti Semin. Mat. Fis. Univ. Modena 35 (1987), 159-165. Zbl 0636.28005, MR 0922998 |
| Reference:
|
[18] Musiał, K.: Topics in the theory of Pettis integration.Rend. Ist. Math. Univ. Trieste 23 (1991), 177-262. Zbl 0798.46042, MR 1248654 |
| Reference:
|
[19] Musiał, K.: Pettis integral.Handbook of Measure Theory. Vol. I. and II. North-Holland, Amsterdam (2002), 531-586 E. Pap. Zbl 1043.28010, MR 1954622, 10.1016/B978-044450263-6/50013-0 |
| Reference:
|
[20] Pfeffer, W. F.: Derivation and Integration.Cambridge Tracts in Mathematics 140. Cambridge University Press, Cambridge (2001). Zbl 0980.26008, MR 1816996, 10.1017/CBO9780511574764 |
| Reference:
|
[21] Schwabik, Š., Ye, G.: Topics in Banach Space Integration.Series in Real Analysis 10. World Scientific, Hackensack (2005). Zbl 1088.28008, MR 2167754, 10.1142/9789812703286 |
| Reference:
|
[22] Talagrand, M.: Pettis integral and measure theory.Mem. Am. Math. Soc. 51 (1984), 224 pages. Zbl 0582.46049, MR 0756174, 10.1090/memo/0307 |
| Reference:
|
[23] Thomson, B. S.: Derivates of interval functions.Mem. Am. Math. Soc. 452 (1991), 96 pages. Zbl 0734.26003, MR 1078198, 10.1090/memo/0452 |
| Reference:
|
[24] Thomson, B. S.: Differentiation.Handbook of Measure Theory. Volume I. and II. North-Holland, Amsterdam (2002), 179-247 E. Pap. Zbl 1028.28001, MR 1954615, 10.1016/B978-044450263-6/50006-3 |
| Reference:
|
[25] Wu, C., Yao, X.: A Riemann-type definition of the Bochner integral.J. Math. Study 27 (1994), 32-36. Zbl 0947.28010, MR 1318255 |
| Reference:
|
[26] Ye, G.: On Henstock-Kurzweil and McShane integrals of Banach space-valued functions.J. Math. Anal. Appl. 330 (2007), 753-765. Zbl 1160.46028, MR 2308405, 10.1016/j.jmaa.2006.08.020 |
| . |
