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Keywords:
McShane integral; Vitali convergence theorem; equi-integrability
McShane integral; Vitali convergence theorem; equi-integrability
Summary:
The McShane integral of functions $f\:I\rightarrow \mathbb{R}$ defined on an $m$-dimensional interval $I$ is considered in the paper. This integral is known to be equivalent to the Lebesgue integral for which the Vitali convergence theorem holds. For McShane integrable sequences of functions a convergence theorem based on the concept of equi-integrability is proved and it is shown that this theorem is equivalent to the Vitali convergence theorem.
References:
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[2] E. J. McShane: A Riemann-type integral that includes Lebesgue-Stieltjes, Bochner and stochastic integrals. Mem. Am. Math. Soc. 88 (1969). MR 0265527 | Zbl 0188.35702
[3] I. P. Natanson: Theory of Functions of a Real Variable. Frederick Ungar, New York, 1955, 1960. MR 0067952
[4] Š. Schwabik, Ye Guoju: On the strong McShane integral of functions with values in a Banach space. Czechoslovak Math. J. 51 (2001), 819–828. DOI 10.1023/A:1013721114330 | MR 1864044
[5] J. Kurzweil, Š. Schwabik: On McShane integrability of Banach space-valued functions. (to appear). MR 2083811
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