On indefinite BV-integrals

In 1986 Bruckner, Fleissner and Foran [2] obtained a descriptive definition of a minimal extension of the Lebesgue integral which integrates the derivative of any differentiable function. Recently, Bongiorno, Di Piazza and Preiss [1] showed that this minimal integral can be obtained from McShane’s definition of the Lebesgue integral [4] by imposing a mild regularity condition on McShane’s partitions. The one-dimensional BV-integral defined in [5, Definition 13.4.2] lies properly in between the Lebesgue and Denjoy-Perron integrals [5, Theorem 11.4.5 and Example 12.3.5], and integrates the derivative (defined almost everywhere) of any function which is pointwise Lipschitz at all but countably many points [5, Theorem 12.2.5]. Moreover, in dimension one, the BV-integral is obtained from McShane’s definition of the Lebesgue integral by using McShane’s partitions consisting of finite unions of compact intervals, and imposing a regularity condition that is only slightly stronger than that employed in [1]. Thus it is natural to ask whether the BV-integral could be the minimal extension of the Lebesgue integral which integrates the derivative of any function that is Lipschitz at all but countably many points. We show by example the answer to this question is negative. By R we denote the set of all real numbers equipped with its usual order and topology. The diameter and Lebesgue measure of a set E ⊂ R are denoted by d(E) and |E|, respectively. Unless specified otherwise, all functions considered in this note are real-valued. A cell is a compact nondegenerate subinterval of R. A finite nonempty union of cells is called a figure. We say figures A and B overlap whenever |A ∩B| > 0. The perimeter of a figure A, denoted by ‖A‖, is the number of the boundary points of A; clearly, ‖A‖ equals twice the number of the connected components