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A concept of absolute continuity and a Riemann type integral

B. Bongiorno, Washek Frank Pfeffer (1992)

Commentationes Mathematicae Universitatis Carolinae

We present a descriptive definition of a multidimensional generalized Riemann integral based on a concept of generalized absolute continuity for additive functions of sets of bounded variation.

A Daniell integral approach to nonstandard Kurzweil-Henstock integral

Ricardo Bianconi, João C. Prandini, Cláudio Possani (1999)

Czechoslovak Mathematical Journal

A workable nonstandard definition of the Kurzweil-Henstock integral is given via a Daniell integral approach. This allows us to study the HL class of functions from . The theory is recovered together with a few new results.

A descriptive definition of a BV integral in the real line

Diana Caponetti, Valeria Marraffa (1999)

Mathematica Bohemica

A descriptive characterization of a Riemann type integral, defined by BV partition of unity, is given and the result is used to prove a version of the controlled convergence theorem.

A full characterization of multipliers for the strong ρ -integral in the euclidean space

Lee Tuo-Yeong (2004)

Czechoslovak Mathematical Journal

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong ρ -integral, introduced by Jarník and Kurzweil. Let ( 𝒮 ρ ( E ) , · ) be the space of all strongly ρ -integrable functions on a multidimensional compact interval E , equipped with the Alexiewicz norm · . We show that each element in the dual space of ( 𝒮 ρ ( E ) , · ) can be represented as a strong ρ -integral. Consequently, we prove that f g is strongly ρ -integrable on E for each strongly ρ -integrable function f if and only if g is almost everywhere...

A full descriptive definition of the BV-integral

B. Bongiorno, Luisa Di Piazza, Washek Frank Pfeffer (1995)

Commentationes Mathematicae Universitatis Carolinae

We present a Cauchy test for the almost derivability of additive functions of bounded BV sets. The test yields a full descriptive definition of a coordinate free Riemann type integral.

A Komlós-type theorem for the set-valued Henstock-Kurzweil-Pettis integral and applications

Bianca Satco (2006)

Czechoslovak Mathematical Journal

This paper presents a Komlós theorem that extends to the case of the set-valued Henstock-Kurzweil-Pettis integral a result obtained by Balder and Hess (in the integrably bounded case) and also a result of Hess and Ziat (in the Pettis integrability setting). As applications, a solution to a best approximation problem is given, weak compactness results are deduced and, finally, an existence theorem for an integral inclusion involving the Henstock-Kurzweil-Pettis set-valued integral is obtained.

A nonexistence result for the Kurzweil integral

Pavel Krejčí, Jaroslav Kurzweil (2002)

Mathematica Bohemica

It is shown that there exist a continuous function f and a regulated function g defined on the interval [ 0 , 1 ] such that g vanishes everywhere except for a countable set, and the K * -integral of f with respect to g does not exist. The problem was motivated by extensions of evolution variational inequalities to the space of regulated functions.

A note on integration by parts for abstract Perron-Stieltjes integrals

Štefan Schwabik (2001)

Mathematica Bohemica

Integration by parts results concerning Stieltjes integrals for functions with values in Banach spaces are presented. The background of the theory is the Kurzweil approach to integration based on Riemann type integral sums, which leads to the Perron integral.

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