On the integration of Riemann-measurable vector-valued functions. (English) Zbl 1379.26013
The article presents various characterizations of Henstock, McShane and other generalizations of Riemann definition of integral of vector valued functions in the framework of Riemann-measurable functions. Most of these characterizations are given following the descriptive approach, namely by studying the properties of the primitives. On the way, the authors clarify important aspects related to different absolute continuity notions. Convergence theorems for Riemann-measurable functions are also obtained, in particular two versions of the dominated convergence theorem.
Reviewer: Bianca-Renata Satco (Suceava)
MSC:
| 26A39 | Denjoy and Perron integrals, other special integrals |
| 28B05 | Vector-valued set functions, measures and integrals |
| 46G10 | Vector-valued measures and integration |
Keywords:
Birkhoff integrals; McShane integrals; Henstock integrals; Pettis integrals; Lebesgue measurable gauge; Riemann-measurable function; almost uniform convergence; bounded variation; \(ACG_{\ast}\) functions; \(ACG_{\delta }^{\ast}\) functionsReferences:
| [1] | Bartle, R.G.: A modern theory of integration. Graduate Studies in Mathematics, Vol. 32, American Mathematical Society, Providence (2001) · Zbl 0968.26001 |
| [2] | Birkhoff, G.: Integration of functions with values in a Banach space. Trans. Am. Math. Soc. 38(2), 357-378 (1935) · JFM 61.0234.01 |
| [3] | Balcerzak, M., Musiał, K.: A convergence theorem for the Birkhoff integral. Funct. Approx. Comment. Math. 50(1), 161-168 (2014) · Zbl 1292.28006 |
| [4] | Balcerzak, M., Potyrała, M.: Convergence theorems for the Birkhoff integral. Czechoslov. Math. J. 58(4), 1207-1219 (2008) · Zbl 1174.28011 · doi:10.1007/s10587-008-0080-1 |
| [5] | Dilworth, S.J., Girardi, M.: Nowhere weak differentiability of the Pettis integral. Quaest. Math. 18(4), 365-380 (1995) · Zbl 0856.28006 · doi:10.1080/16073606.1995.9631809 |
| [6] | Di Piazza, L., Marraffa, V.: An equivalent definition of the vector-valued McShane integral by means of partitions of unity. Stud. Math. 151(2), 175-185 (2002) · Zbl 1005.28009 · doi:10.4064/sm151-2-5 |
| [7] | Di Piazza, L., Marraffa, V.: The McShane, PU and Henstock integrals of Banach valued functions. Czechoslov. Math. J. 52(3), 609-633 (2002) · Zbl 1011.28007 · doi:10.1023/A:1021736031567 |
| [8] | Fremlin, D.H.: The Henstock and McShane integrals of vector-valued functions. Ill. J. Math. 38(3), 471-479 (1994) · Zbl 0797.28006 |
| [9] | Fremlin, D.H.: The generalized McShane integral. Ill. J. Math. 39(1), 39-67 (1995) · Zbl 0810.28006 |
| [10] | Fremlin, D.H.: The McShane and Birkhoff integrals of vector-valued functions, University of Essex Mathematics Department Research Report 92-10, version of 18.5.07 available at http://www.essex.ac.uk/maths/people/fremlin/preprints.htm · Zbl 0797.28006 |
| [11] | Fremlin, D.H., Mendoza, J.: On the integration of vector-valued functions. Ill. J. Math. 38(1), 127-147 (1994) · Zbl 0790.28004 |
| [12] | Gordon, R.A.: Integration and differentiation in a Banach space. Ph.D. Thesis, University of Illinois at Urbana-Champaign (1987) |
| [13] | Gordon, R.A.: The Denjoy extension of the Bochner, Pettis, and Dunford integrals. Stud. Math. 92(1), 73-91 (1989) · Zbl 0681.28006 |
| [14] | Gordon, R.A.: Another look at a convergence theorem for the Henstock integral. Real Anal. Exch. 15 (1989/90), 724-728 · Zbl 0708.26005 |
| [15] | Gordon, R.A.: The McShane integral of Banach-valued functions. Ill. J. Math. 34(3), 557-567 (1990) · Zbl 0685.28003 |
| [16] | Gordon, R.: Riemann integration in Banach spaces. Rocky Mountain J. Math. 21(3), 923-949 (1991) · Zbl 0764.28008 · doi:10.1216/rmjm/1181072923 |
| [17] | Gordon, R.A.: The integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics, Vol. 4. American Mathematical Society, Providence RI (1994) · Zbl 0807.26004 |
| [18] | Kadets, V.M., Tseytlin, L.M.: On “integration” of non-integrable vector-valued functions. Mat. Fiz. Anal. Geom. 7(1), 49-65 (2000) · Zbl 0974.28007 |
| [19] | Kadets, V., Shumyatskiy, B., Shvidkoy, R., Tseytlin, L., Zheltukhin, K.: Some remarks on vector-valued integration. Mat. Fiz. Anal. Geom. 9(1), 48-65 (2002) · Zbl 1084.28008 |
| [20] | Kolmogoroff, A.: Untersuchungen über den Integralbegriff. (German). Math. Ann. 103(1), 654-696 (1930) · JFM 56.0923.01 · doi:10.1007/BF01455714 |
| [21] | Kurzweil, J., Schwabik, S̆.: On McShane integrability of Banach space-valued functions. Real Anal. Exch. 29(2003/04), 763-780 · Zbl 1078.28007 |
| [22] | Lu, S.P., Lee, P.Y.: Globally small Riemann sums and the Henstock integral. Real Anal. Exch. 16(1990/91), 537-545 · Zbl 0757.26011 |
| [23] | Marraffa, V.: A descriptive characterization of the variational Henstock integral. In: Proceedings of the International Mathematics Conference (Manila, 1998). Matimyás Mat. 22(1999), 73-84 · Zbl 1030.28005 |
| [24] | Marraffa, V.: A Birkhoff type integral and the Bourgain property in a locally convex space. Real Anal. Exch. 32(2), 409-427 (2007) · Zbl 1142.28011 |
| [25] | Naralenkov, K.M.: Asymptotic structure of Banach spaces and Riemann integration. Real Anal. Exch. 33(2007/08), 111-124 · Zbl 1151.26008 |
| [26] | Naralenkov, K.: On Denjoy type extensions of the Pettis integral. Czechoslov. Math. J. 60(3), 737-750 (2010) · Zbl 1224.26028 · doi:10.1007/s10587-010-0047-x |
| [27] | Naralenkov, K.: Several comments on the Henstock-Kurzweil and McShane integrals of vector-valued functions. Czechoslov. Math. J. 61(4), 1091-1106 (2011) · Zbl 1249.26010 · doi:10.1007/s10587-011-0050-x |
| [28] | Naralenkov, K.M.: A Henstock-Kurzweil integrable vector-valued function which is not McShane integrable on any portion. Quaest. Math. 35(1), 11-21 (2012) · Zbl 1274.26015 · doi:10.2989/16073606.2012.671160 |
| [29] | Naralenkov, K.M.: A Lusin type measurability property for vector-valued functions. J. Math. Anal. Appl. 417(1), 293-307 (2014) · Zbl 1305.28025 · doi:10.1016/j.jmaa.2014.03.029 |
| [30] | Naralenkov, K.M.: Some comments on scalar differentiation of vector-valued functions. Bull. Aust. Math. Soc. 91(2), 311-321 (2015) · Zbl 1326.46038 · doi:10.1017/S0004972714000823 |
| [31] | Pettis, B.J.: On integration in vector spaces. Trans. Am. Math. Soc. 44(2), 277-304 (1938) · Zbl 0019.41603 · doi:10.1090/S0002-9947-1938-1501970-8 |
| [32] | Potyrała, M.: Some remarks about Birkhoff and Riemann-Lebesgue integrability of vector valued functions. Tatra Mt. Math. Publ. 35, 97-106 (2007) · Zbl 1164.28010 |
| [33] | Rodríguez, J.: On the existence of Pettis integrable functions which are not Birkhoff integrable. Proc. Am. Math. Soc. 133(4), 1157-1163 (2005) · Zbl 1058.28010 · doi:10.1090/S0002-9939-04-07665-8 |
| [34] | Rodríguez, J.: Pointwise limits of Birkhoff integrable functions. Proc. Am. Math. Soc. 137(1), 235-245 (2009) · Zbl 1185.28018 · doi:10.1090/S0002-9939-08-09589-0 |
| [35] | Rodríguez, J.: Convergence theorems for the Birkhoff integral. Houst. J. Math. 35(2), 541-551 (2009) · Zbl 1175.28008 |
| [36] | Snow, D.O.: On integration of vector-valued functions. Can. J. Math. 10, 399-412 (1958) · Zbl 0082.11003 · doi:10.4153/CJM-1958-039-9 |
| [37] | Solodov, A.P.: On the limits of the generalization of the Kolmogorov integral. (in Russian) Mat. Zametki 77(2005), 258-272; translation in Math. Notes 77 (2005), no. 1-2, 232-245 · Zbl 1073.28007 |
| [38] | Swartz, C.: Norm convergence and uniform integrability for the Henstock-Kurzweil integral. Real Anal. Exch. 24(1998/99), no. 1, 423-426 · Zbl 0943.26022 |
| [39] | Talagrand, M.: Pettis integral and measure theory. Mem. Am. Math. Soc. 51 (1984), No. 307 · Zbl 0582.46049 |
| [40] | Wang, P.: Equi-integrability and controlled convergence for the Henstock integral. Real Anal. Exch. 19(1993/94), 236-241 · Zbl 0803.26006 |
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