Henstock-Kurzweil integral on BV sets

Jan Malý; Washek Frank Pfeffer

Mathematica Bohemica (2016)

  • Volume: 141, Issue: 2, page 217-237
  • ISSN: 0862-7959

Abstract

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The generalized Riemann integral of Pfeffer (1991) is defined on all bounded BV subsets of ℝ n , but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of σ -finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of BV sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect to the formation of improper integrals. Its definition in ℝ coincides with the Henstock-Kurzweil definition of the Denjoy-Perron integral.

How to cite

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Malý, Jan, and Pfeffer, Washek Frank. "Henstock-Kurzweil integral on ${\rm BV}$ sets." Mathematica Bohemica 141.2 (2016): 217-237. <http://eudml.org/doc/276988>.

@article{Malý2016,
abstract = {The generalized Riemann integral of Pfeffer (1991) is defined on all bounded $\rm BV$ subsets of $\mathbb \{R\}^n$, but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of $\sigma $-finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of $\rm BV$ sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect to the formation of improper integrals. Its definition in $\mathbb \{R\}$ coincides with the Henstock-Kurzweil definition of the Denjoy-Perron integral.},
author = {Malý, Jan, Pfeffer, Washek Frank},
journal = {Mathematica Bohemica},
keywords = {Henstock-Kurzweil integral; charge; $\rm BV$ set},
language = {eng},
number = {2},
pages = {217-237},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Henstock-Kurzweil integral on $\{\rm BV\}$ sets},
url = {http://eudml.org/doc/276988},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Malý, Jan
AU - Pfeffer, Washek Frank
TI - Henstock-Kurzweil integral on ${\rm BV}$ sets
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 2
SP - 217
EP - 237
AB - The generalized Riemann integral of Pfeffer (1991) is defined on all bounded $\rm BV$ subsets of $\mathbb {R}^n$, but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of $\sigma $-finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of $\rm BV$ sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect to the formation of improper integrals. Its definition in $\mathbb {R}$ coincides with the Henstock-Kurzweil definition of the Denjoy-Perron integral.
LA - eng
KW - Henstock-Kurzweil integral; charge; $\rm BV$ set
UR - http://eudml.org/doc/276988
ER -

References

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