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Absolute continuity with respect to a subset of an interval

Lucie Loukotová (2017)

Commentationes Mathematicae Universitatis Carolinae

The aim of this paper is to introduce a generalization of the classical absolute continuity to a relative case, with respect to a subset M of an interval I . This generalization is based on adding more requirements to disjoint systems { ( a k , b k ) } K from the classical definition of absolute continuity – these systems should be not too far from M and should be small relative to some covers of M . We discuss basic properties of relative absolutely continuous functions and compare this class with other classes of...

Riemann-type definition of the improper integrals

Donatella Bongiorno (2004)

Czechoslovak Mathematical Journal

Riemann-type definitions of the Riemann improper integral and of the Lebesgue improper integral are obtained from McShane’s definition of the Lebesgue integral by imposing a Kurzweil-Henstock’s condition on McShane’s partitions.

Role of the Harnack extension principle in the Kurzweil-Stieltjes integral

Umi Mahnuna Hanung (2024)

Mathematica Bohemica

In the theories of integration and of ordinary differential and integral equations, convergence theorems provide one of the most widely used tools. Since the values of the Kurzweil-Stieltjes integrals over various kinds of bounded intervals having the same infimum and supremum need not coincide, the Harnack extension principle in the Kurzweil-Henstock integral, which is a key step to supply convergence theorems, cannot be easily extended to the Kurzweil-type Stieltjes integrals with discontinuous...

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