zbMATH — the first resource for mathematics

The Vitali convergence theorem for the vector-valued McShane integral. (English) Zbl 1051.28007
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.
MSC:
 28B05 Vector-valued set functions, measures and integrals 46G10 Vector-valued measures and integration 26A39 Denjoy and Perron integrals, other special integrals
Full Text: