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The Ward, Perron and Henstock-Kurzweil integrals with respect to abstract derivation bases in Riesz spaces. (English) Zbl 1114.26009

Extensions of three classical concepts of integration (Ward, Perron and Henstock-Kurzweil) to the Riesz-space-valued functions are considered. First, the extensions of the Ward and Perron integrals with respect to a given derivation basis are defined. While the definition of the Ward integral (as well as the Kurzweil-Henstock one) is a straightforward generalization of the respective definition for real functions, the situation with the Perron integral is more delicate. The most natural extension does not yield the desired result. To this end, derivatives of another type, the so-called \(g\)-derivatives, must be used to obtain a suitably defined Perron integral. In this case all three concepts of integral are shown to be equivalent.

MSC:

26A39 Denjoy and Perron integrals, other special integrals
28A15 Abstract differentiation theory, differentiation of set functions
28B05 Vector-valued set functions, measures and integrals
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
46G10 Vector-valued measures and integration
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