Henstock-Kurzweil integral on BV sets. (Q2810144)

Starting in the early nineteen eighties, various extensions of the Kurzweil-Henstock integral in \(\mathbb{R}^n\) have been introduced, which can integrate divergences of differentiable vector fields on various classes of bounded subset of \(\mathbb{R}^n\) and provide a divergence theorem. All those extensions require that the subsets of the partitions associated to the Riemann sums in the definition satisfy a suitable regularity condition. For example, in the simple case of a parallelotope, the ratio between its larger and its smaller side must be controlled.NEWLINENEWLINE The present paper is a further striking contribution to this problem. By introducing a new stricter regularity condition on the partitioning sets, the authors provide a new integral on bounded BV sets which has better additivity properties than Pfeffer's R-integral, is finitely additive and coincides with the Denjoy-Perron integral in dimension one. This new integral is furthermore lipeomorphism-invariant and closed with respect to the formation of improper integrals.