Real Analysis Exchange

On McShane integrability of Banach space-valued functions.

Jaroslav Kurzweil and Štefan Schwabik

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Abstract

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

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

Dates
First available in Project Euclid: 7 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149698538

Mathematical Reviews number (MathSciNet)
MR2083811

Zentralblatt MATH identifier
1078.28007

Subjects
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

Keywords
McShane integral vector integration

Citation

Kurzweil, Jaroslav; Schwabik, Štefan. On McShane integrability of Banach space-valued functions. Real Anal. Exchange 29 (2003), no. 2, 763--780. https://projecteuclid.org/euclid.rae/1149698538


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References

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  • D. H. Fremlin, The generalized McShane integral, Illinois Journal of Math., 39 (1995), 30–67.
  • R. A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, American Mathematical Society, 1994.