# Article

Full entry | PDF   (0.3 MB)
Keywords:
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:
[1] R. A. Gordon: The integrals of Lebesgue, Denjoy, Perron, and Henstock. American Mathematical Society, Providence, RI, 1994. MR 1288751 | Zbl 0807.26004
[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. MR 1864044
[5] J. Kurzweil, Š. Schwabik: On McShane integrability of Banach space-valued functions. (to appear). MR 2083811

Partner of