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Keywords:
Kurzweil-Henstock integral; Kurzweil-Henstock-Pettis integral; Pettis integral
Summary:
We study the integrability of Banach valued strongly measurable functions defined on $[0,1]$. In case of functions $f$ given by $\sum _{n=1}^{\infty } x_n\chi _{E_n}$, where $x_n $ belong to a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for the Bochner and for the Pettis integrability of $f$ (cf Musial (1991)). In this paper we give some conditions for the Kurzweil-Henstock and the Kurzweil-Henstock-Pettis integrability of such functions.
References:
[1] A. Alexiewicz: Linear functionals on Denjoy integrable functions. Coll. Math. 1 (1948), 289–293. MR 0030120 | Zbl 0037.32302
[2] B. Bongiorno: The Henstock-Kurzweil integral. Handbook of Measure Theory I, E. Pap ed., Elsevier Amsterdam, 2002, pp. 587–615. MR 1954623 | Zbl 1024.26004
[3] J. Diestel, J. J. Uhl: Vector measures. Math. Surveys, vol. 15, AMS, Providence, R.I., 1977. MR 0453964
[4] D. H. J. Fremlin: The Henstock and McShane integrals of vector-valued functions. Illinois J. Math. 38 (1994), 471–479. MR 1269699 | Zbl 0797.28006
[5] J. L. Gamez, J. Mendoza: On Denjoy-Dunford and Denjoy-Pettis integrals. Studia Math. 130 (1998), 115–133. MR 1623348
[6] P. Y. Lee: Lanzhou Lectures on Henstock Integration. World Scientific, Singapore, 1989. MR 1050957 | Zbl 0699.26004
[7] K. Musial: Topics in the theory of Pettis integration. Rend. Istit. Mat. Univ. Trieste 23 (1991), 177–262. MR 1248654 | Zbl 0798.46042
[8] C. Swartz: Norm convergence and uniform integrability for the Kurzweil-Henstock integral. Real Anal. Exchange 24 (1998/99), 423–426. MR 1691761
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