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McShane equi-integrability and Vitali’s convergence theorem. (English) Zbl 1051.26012

Summary: The McShane integral of functions \(f\: I \to \mathbb R\) 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.

MSC:

26B99 Functions of several variables
26A39 Denjoy and Perron integrals, other special integrals
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