Geodesic
The Vitali convergence theorem for the vector-valued McShane integral
Mathematica Bohemica, Tome 129 (2004) no. 2, pp. 159-176

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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.
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.
DOI : 10.21136/MB.2004.133906
Classification : 26A39, 26A42, 28B05, 46G10
Keywords: vector-valued McShane integral; Vitali theorem; norm convergence 
Reynolds, Richard; Swartz, Charles. The Vitali convergence theorem for the vector-valued McShane integral. Mathematica Bohemica, Tome 129 (2004) no. 2, pp. 159-176. doi: 10.21136/MB.2004.133906
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