Real Analysis Exchange

On McShane integrability of Banach space-valued functions.

Jaroslav Kurzweil and Štefan Schwabik

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The McShane integral of Banach space-valued functions $f:I\to X$ defined on an $m$-dimensional interval $I$ is considered in this paper. We show that a McShane integrable function is integrable over measurable sets contained in $I$ (Theorem 9). A certain type of absolute continuity of the indefinite McShane integral with respect to Lebesgue measure is derived (Theorem 11) and we show that the indefinite McShane integral is countably additive (Theorem 16). Allowing more general partitions using measurable sets instead of intervals another McShane type integral is defined and we show that it is equivalent to the original McShane integral (Theorem 21)

Article information

Real Anal. Exchange, Volume 29, Number 2 (2003), 763-780.

First available in Project Euclid: 7 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10] 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22] 26A39: Denjoy and Perron integrals, other special integrals

McShane integral vector integration


Kurzweil, Jaroslav; Schwabik, Štefan. On McShane integrability of Banach space-valued functions. Real Anal. Exchange 29 (2003), no. 2, 763--780.

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