Henstock-Kurzweil integral on ${\rm BV}$ sets

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.}

Keywords

• Henstock-Kurzweil integral
• charge
• ${\rm BV$ set}