Real Analysis Exchange

A Generalized Riemann Integral for Banach-Valued Functions

Jean-Christophe Feauveau

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We shall develop the properties of an integral for Banach-valued functions. The formalism is the generalized Riemann integral introduced by Kurzweil \cite{Kur} and Henstock \cite{Hen}. More precisely, the presentation is close to the McShane approach \cite{McS}. Besides its simplicity of presentation, four advantages characterize this theory: %{\leftskip=1cm \item{(i)} the definition can be used for real-valued functions, and can be generalized without modification to general real and complex Banach spaces; \item{(ii)} when a function is integrable its norm is also integrable, and the proof is straightforward from the definition; \item{(iii)} for finite dimension spaces the theory is equivalent to the McShane's theory, which is itself equivalent to the Lebesgue's theory; \item{(iv)} and lastly, for general Banach space, we can prove the equivalence to the Bochner's theory. \par%}

Article information

Real Anal. Exchange, Volume 25, Number 2 (1999), 919-930.

First available in Project Euclid: 3 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

McShane integral Bochner integral vector-valued integral


Feauveau, Jean-Christophe. A Generalized Riemann Integral for Banach-Valued Functions. Real Anal. Exchange 25 (1999), no. 2, 919--930.

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