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Relations among gauge and Pettis integrals for \(cwk(X)\)-valued multifunctions. (English) Zbl 1400.28020

The authors investigate integrable multifunctions whose values are weakly compact and convex subsets of a general Banach space (not necessarily separable). The three decomposition theorems presented in the paper account for the relation between the so-called ‘gauge integrals’ (Henstock, McShane, Birkhoff) and the Pettis integral of multifunctions. These results are then applied to deduce characterizations of McShane and Birkhoff integrability.
Concerning the variational Henstock integral, the paper’s contributions lie in two main points: a complete answer to the Question 3.11 proposed in [D. Candeloro et al., J. Math. Anal. Appl. 441, No. 1, 293–308 (2016; Zbl 1339.28016)]; an alternative proof of a decomposition result delivered by the authors in [“Some new results on integration for multifunction”, Preprint, arXiv:1610.09151]. Moreover, it is shown that the variational Henstock integral coincides with its stronger version, namely, with the variational \(\mathcal{H}\)-integral.

MSC:

28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
26A39 Denjoy and Perron integrals, other special integrals
26E25 Set-valued functions
28B05 Vector-valued set functions, measures and integrals
46G10 Vector-valued measures and integration
54C60 Set-valued maps in general topology
54C65 Selections in general topology

Citations:

Zbl 1339.28016
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References:

[1] Aviles, A., Plebanek, G., Rodríguez, J.: The McShane integral in weakly compactly generated spaces. J. Funct. Anal. 259(11), 2776-2792 (2010) · Zbl 1213.46037 · doi:10.1016/j.jfa.2010.08.007
[2] Boccuto, A., Candeloro, D., Sambucini, A.R.: Henstock multivalued integrability in Banach lattices with respect to pointwise non atomic measures. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26(4), 363-383 (2015). doi:10.4171/RLM/710 · Zbl 1329.28024 · doi:10.4171/RLM/710
[3] Boccuto, A., Sambucini, A.R.: A note on comparison between Birkhoff and McShane-type integrals for multifunctions. Real Anal. Exch. 37(2), 315-324 (2012) · Zbl 1277.28016 · doi:10.14321/realanalexch.37.2.0315
[4] Bongiorno, B., Di Piazza, L., Musiał, K.: A variational Henstock integral characterization of the Radon-Nikodym Property. Ill. J. Math. 53(1), 87-99 (2009) · Zbl 1200.46021
[5] Bongiorno, B., Di Piazza, L., Skvortsov, V.: A new full descriptive characterization of Denjoy-Perron integral. Real Anal. Exch. 21, 256-263 (1995/96) · Zbl 0879.26026
[6] Candeloro, D., Di Piazza, L., Musiał, K., Sambucini, A.R.: Gauge integrals and selections of weakly compact valued multifunctions. J. Math. Anal. Appl. 441(1), 293-308 (2016). doi:10.1016/j.jmaa.2016.04.009 · Zbl 1339.28016 · doi:10.1016/j.jmaa.2016.04.009
[7] Candeloro, D., Di Piazza, L., Musiał, K., Sambucini, A.R.: Some new results on integration for multifunction. arXiv:1610.09151 (2016) · Zbl 1400.28020
[8] Candeloro, D., Sambucini, A.R.: Order-type Henstock and Mc Shane integrals in Banach lattices setting. In: Sisy 2014 - IEEE 12th International Symposium on Intelligent Systems and Informatics, Subotica (2014) doi:10.1109/SISY.2014.6923557 · Zbl 0790.28004
[9] Candeloro, D., Sambucini, A.R.: Comparison between some norm and order gauge integrals in Banach lattices. Panam. Math. J. 25(3), 1-16 (2015). arXiv:1503.04968 [math.FA] · Zbl 1337.28021
[10] Caponetti, D., Marraffa, V., Naralenkov, K.: On the integration of Riemann-measurable vector-valued functions. Monatsh. Math. (2016). doi:10.1007/s00605-016-0923-z · Zbl 1379.26013 · doi:10.1007/s00605-016-0923-z
[11] Cascales, B., Rodríguez, J.: Birkhoff integral for multi-valued functions. J. Math. Anal. Appl. 297(2), 540-560 (2004) · Zbl 1066.46037 · doi:10.1016/j.jmaa.2004.03.026
[12] Cascales, C., Kadets, V., Rodríguez, J.: The Pettis integral for multi-valued functions via single-valued ones. J. Math. Anal. Appl. 332(1), 1-10 (2007) · Zbl 1119.28009 · doi:10.1016/j.jmaa.2006.10.003
[13] Cascales, C., Kadets, V., Rodríguez, J.: Measurable selectors and set-valued Pettis integral in non-separable Banach spaces. J. Funct. Anal. 256(3), 673-699 (2009) · Zbl 1160.28004 · doi:10.1016/j.jfa.2008.10.022
[14] Di Piazza, L.: Variational measures in the theory of the integration in \[R^m\] Rm. Czechoslov. Math. J. 51(1), 95-110 (2001) · Zbl 1079.28500 · doi:10.1023/A:1013705821657
[15] Di Piazza, L., Marraffa, V.: The McShane, PU and Henstock integrals of Banach valued functions. Czechoslov. Math. J. 52(127)(3), 609-633 (2002). ISSN: 0011-4642 · Zbl 1011.28007 · doi:10.1023/A:1021736031567
[16] Di Piazza, L., Musiał, K.: A characterization of variationally McShane integrable Banach-space valued functions. Ill. J. Math. 45(1), 279-289 (2001) · Zbl 0999.28006
[17] Di Piazza, L., Musiał, K.: Set-Valued Henstock-Kurzweil-Pettis Integral. Set-Valued Anal. 13, 167-179 (2005) · Zbl 1100.28008 · doi:10.1007/s11228-004-0934-0
[18] Piazza, L.; Musiał, K.; Curbera, GP (ed.); Mockenhaupt, G. (ed.); Ricker, WJ (ed.), A decomposition of Henstock-Kurzweil-Pettis integrable multifunctions, Vector measures, integration and related topics, No. 201, 171-182 (2010), Basel · Zbl 1248.28019
[19] Di Piazza, L., Musiał, K.: Relations among Henstock, McShane and Pettis integrals for multifunctions with compact convex values. Monatsh. Math. 173(4), 459-470 (2014) · Zbl 1293.28006 · doi:10.1007/s00605-013-0594-y
[20] Di Piazza, L., Porcello, G.: Radon-Nikodym theorems for finitely additive multimeasures. Z. Anal. ihre. Anwend. (ZAA) 34(4), 373-389 (2015). doi:10.4171/ZAA/1545 · Zbl 1331.28026 · doi:10.4171/ZAA/1545
[21] Fremlin, D.H.: The McShane and Birkhoff integrals of vector-valued functions. University of Essex Mathematics Department Research Report 92-10 (1999) · Zbl 1277.28016
[22] Fremlin, D.H.: The Henstock and McShane integrals of vector-valued functions. Ill. J. Math. 38(3), 471-479 (1994) · Zbl 0797.28006
[23] Fremlin, D.H., Mendoza, J.: On the integration of vector-valued functions. Ill. J. Math. 38, 127-147 (1994) · Zbl 0790.28004
[24] Fremlin, D.H.: Measure Theory, volume 4: Topological Measure Spaces. Torres Fremlin, Colchester (2003) · Zbl 1166.28001
[25] Gordon, R.A.: The Denjoy extension of the Bochner, Pettis, and Dunford integrals. Stud. Math. 92, 73-91 (1989) · Zbl 0681.28006 · doi:10.4064/sm-92-1-73-91
[26] Gordon, R.A.: The Integrals of Lebesgue, Denjoy, Perron and Henstock, vol. 4. AMS, Providence (1994). (Grad. Stud. Math. ) · Zbl 0807.26004
[27] Himmelberg, C.J., Van Vleck, F.S., Prikry, K.: The Hausdorff metric and measurable selections. Topol. Appl. 20(2), 121-133 (1985) · Zbl 0586.28009 · doi:10.1016/0166-8641(85)90072-0
[28] Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis I and II, : Mathematics and Its Applications, vol. 419. Kluwer Academic Publisher, Dordrecht (1997) · Zbl 0887.47001 · doi:10.1007/978-1-4615-6359-4
[29] Kaliaj, S.B.: Descriptive characterizations of Pettis and strongly McShane integrals. Real Anal. Exch. 40(1), 227-238 (2015) · Zbl 1298.28025 · doi:10.14321/realanalexch.40.1.0227
[30] Labuschagne, C.C.A., Pinchuck, A.L., van Alten, C.J.: A vector lattice version of Rådström’s embedding theorem. Quaest. Math. 30(3), 285-308 (2007) · Zbl 1144.06009 · doi:10.2989/16073600709486200
[31] Malý, J.: Non-absolutely convergent integrals with respect to distributions. Ann. Mat. Pura Appl. 193(5), 1457-1484 (2014) · Zbl 1304.26006 · doi:10.1007/s10231-013-0338-6
[32] Malý, J., Pfeffer, W.F.: Henstock-Kurzweil integral on BV sets. Math. Bohem. 141(2), 217-237 (2016) · Zbl 1413.26026 · doi:10.21136/MB.2016.16
[33] Marraffa, V.: Strongly measurable Kurzweil-Henstock type integrable functions and series. Quaest. Math. 31(4), 379-386 (2008) · Zbl 1177.28030 · doi:10.2989/QM.2008.31.4.6.610
[34] Marraffa, V.: The variational McShane integral in locally convex spaces. Rocky Mt. J. Math. 39(6), 1993-2013 (2009) · Zbl 1187.28019 · doi:10.1216/RMJ-2009-39-6-1993
[35] Musiał, K.: Topics in the theory of Pettis integration. Rend. Istit. Mat. Univ. Trieste 23, 177-262 (1991) · Zbl 0798.46042
[36] Musiał, K.: Pettis integral. In: Pap, E. (ed.) Handbook of Measure Theory I. Elsevier, Amsterdam (2002) · Zbl 1043.28010
[37] Musiał, K.: Pettis integrability of multifunctions with values in arbitrary Banach spaces. J. Convex Anal. 18(3), 769-810 (2011) · Zbl 1245.28011
[38] Musiał, K.: Approximation of Pettis integrable multifunctions with values in arbitrary Banach spaces. J. Convex Anal. 20(3), 833-870 (2013) · Zbl 1284.28006
[39] Naralenkov, K.M.: A Lusin type measurability property for vector-valued functions. J. Math. Anal. Appl. 417(1), 293-307 (2014). doi:10.1016/j.jmaa.2014.03.029 · Zbl 1305.28025 · doi:10.1016/j.jmaa.2014.03.029
[40] Porcello, G.: ‘Multimeasures and integration of multifunctions in Banach spaces’, Dottorato di Ricerca in Matematica e Informatica XXIV ciclo, University of Palermo (Italy) https://iris.unipa.it/retrieve/handle/10447/91026/99048/TesiDottoratoGiovanniPorcello.pdf · Zbl 1302.26004
[41] Solodov, A.P.: On the limits of the generalization of the Kolmogorov integral. Mat. Zamet. 77(2), 258-272 (2005) · Zbl 1073.28007 · doi:10.4213/mzm2488
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