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A generalized Riemann integral for Banach-valued functions. (English) Zbl 1022.28004

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

MSC:

28B05 Vector-valued set functions, measures and integrals
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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