EUDML

We study the relationship between derivates and variational measures of additive functions defined on families of figures or bounded sets of finite perimeter. Our results, valid in all dimensions, include a generalization of Ward’s theorem, a necessary and sufficient condition for derivability, and full descriptive definitions of certain conditionally convergent integrals.

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.

Hake's property of a multidimensional generalized Riemann integral

Jean Mawhin; Washek Frank Pfeffer — 1990

Czechoslovak Mathematical Journal

Henstock-Kurzweil integral on $BV$ sets

Jan Malý; Washek Frank Pfeffer — 2016

Mathematica Bohemica

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...

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.

On indefinite BV-integrals

Donatella Bongiorno; Udayan B. Darji; Washek Frank Pfeffer — 2000

Commentationes Mathematicae Universitatis Carolinae

We present an example of a locally BV-integrable function in the real line whose indefinite integral is not the sum of a locally absolutely continuous function and a function that is Lipschitz at all but countably many points.