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MATHEMATICA BOHEMICA, Vol. 129, No. 2, pp. 159-176 (2004)
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The Vitali convergence theorem for the vector-valued McShane integral

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Richard Reynolds, Charles Swartz

* Richard Reynolds*, Department of Mathematics and Statistics, P.O. Box 871804, Arizona State University, Tempe, AZ 85287-1804, USA, e-mail: ` rich@math.asu.edu`; * Charles Swartz*, New Mexico State University, Department of Mathematical Sciences, P.O. Box 30001 Department 3MB, Las Cruces, New Mexico 88003-8001, USA, e-mail: ` cswartz@nmsu.edu`

**Abstract:** 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 $\Bbb R^{n}$ 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.

**Keywords:** vector-valued McShane integral, Vitali theorem

**Classification (MSC2000):** 28B05, 46G10, 26A39

**Full text of the article:**

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