# Article

Full entry | PDF   (0.3 MB)
Keywords:
vector-valued McShane integral; Vitali theorem; norm convergence
Summary:
The classical Vitali convergence theorem gives necessary and sufficient conditions for norm convergence in the space of Lebesgue integrable functions. Although there are versions of the Vitali convergence theorem for the vector valued McShane and Pettis integrals given by Fremlin and Mendoza, these results do not involve norm convergence in the respective spaces. There is a version of the Vitali convergence theorem for scalar valued functions defined on compact intervals in \$\mathbb{R}^{n}\$ given by Kurzweil and Schwabik, but again this version does not consider norm convergence in the space of integrable functions. In this paper we give a version of the Vitali convergence theorem for norm convergence in the space of vector-valued McShane integrable functions.
References:
[1] N. Dunford, J. Schwartz: Linear Operators. Interscience, N.Y., 1958.
[2] D. H. Fremlin, J. Mendoza: On the integration of vector-valued functions. Illinois J. Math. 38 (1994), 127–147. DOI 10.1215/ijm/1255986891 | MR 1245838
[3] D. H. Fremlin: The generalized McShane integral. Illinois J. Math. 39 (1995), 39–67. DOI 10.1215/ijm/1255986628 | MR 1299648 | Zbl 0810.28006
[4] R. Gordon: The McShane integral of Banach-valued functions. Illinois J. Math. 34 (1990), 557–567. DOI 10.1215/ijm/1255988170 | MR 1053562 | Zbl 0685.28003
[5] R. Gordon: Some comments on the McShane and Henstock integrals. Real Anal. Exchange 23 (1997/98), 329–341. MR 1609917
[6] E. Hewitt, K. Stromberg: Real and Abstract Analysis. Springer, N.Y., 1965. MR 0367121
[7] J. Kurzweil, Š. Schwabik: On the McShane integrability of Banach space-valued functions. (to appear). MR 2083811
[8] J. Kurzweil, Š. Schwabik: McShane equi-integrability and Vitali’s convergence theorem. Math. Bohem. 129 (2004), 141–157. MR 2073511
[9] E. J. McShane: Unified Integration. Academic Press, N.Y., 1983. MR 0740710 | Zbl 0551.28001
[10] K. Musial: Topics in the theory of Pettis integration. Rendiconti Inst. Mat. Univ. Trieste 23 (1991), 177–262. MR 1248654 | Zbl 0798.46042
[11] R. Reynolds: The Generalized McShane Integral for Vector-Valued Functions. Ph.D. dissertation, New Mexico State University, 1997.
[12] H. Royden: Real Analysis. Macmillan, N.Y., 1988. MR 0151555 | Zbl 0704.26006
[13] C. Swartz: Measure, Integration, and Function Spaces. World Scientific, Singapore, 1994. MR 1337502 | Zbl 0814.28001
[14] C. Swartz: Beppo Levi’s theorem for the vector-valued McShane integral. Bull. Belgian Math. Soc. 4 (1997), 589–599. DOI 10.36045/bbms/1105737762 | MR 1600292 | Zbl 1038.46505
[15] C. Swartz: Uniform integrability and mean convergence for the vector-valued McShane integral. Real Anal. Exchange 23 (1997/98), 303–312. MR 1609766
[16] C. Swartz: Introduction to Gauge Integrals. World Scientific, Singapore, 2001. MR 1845270 | Zbl 0982.26006

Partner of