The BV-integral, introduced in [5, Definition 5.1] under the name “variational integral”, is a coordinate free generalization of the Lebesgue integral defined on all bounded Caccioppoli sets. Unlike the Lebesgue integral, it integrates partial derivatives of differentiable functions and provides the unrestricted Gauss-Green theorem. The purpose of this note is to present a complete characterization of those additive functions of bounded Caccioppoli sets that are indefinite BVintegrals (Theorem 3.9).

We define a well-behaved multidimensional Riemann type integral such that the divergence of any vector field continuous in a compact interval and differentiable in its interior is integrable, and the integral equals the flux of the vector field out of the interval.

In the context of Lebesgue integration the Gauss-Green theorem is proved for bounded vector fields with substantial sets of singularities with respect to continuity and differentiability. The resulting integration by parts is applied to removable sets for the Cauchy-Riemann, Laplace, and minimal surface equations. A simple connection between the Gauss-Green theorem and distributional divergence is established.

Applying a very general Gauss–Green theorem established for the generalized Riemann integral, we obtain simple proofs of new results about removable sets of singularities for the Laplace and minimal surface equations. We treat simultaneously singularities with respect to differentiability and continuity.

A charge in the Euclidean space R-m is an additive function defined on the family of all bounded BV sets equipped with a suitable topology. We define derivatives of charges and show that each measurable function defined on R-m is equal almost everywhere to the derivative of a charge.

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.

For an invariant generalized Riemann integral in R-m, we obtain the following results. (1) A function is a multiplier for the space of locally integrable functions if and only if it is locally bounded and locally BV. (2) The dual of the space of all functions integrable in a bounded BV set A is linearly isomorphic to the space of all bounded BV functions vanishing outside A, and each element of the dual has the usual integral representation. (3) On Lipschitz domains an integration by parts formula holds for any continuous function that is pointwise Lipschitz, everywhere except on a set of sigma-finite (m - 1)-dimensional Hausdorff measure.

We show that the variational measure associated with an additive continuous function of bounded BV sets is the same, irrespective whether it is defined by means of bounded BV sets or by means of finite unions of compact intervals. An application of this result to multidimensional Riemann type integrals is given.

We consider a specific Riemann type integral, called the gage integral. Using variational measures, we characterize all additive functions of intervals that are indefinite gage integrals. The characterization generalizes the descriptive definition of the classical Denjoy-Perron integral to all dimensions.

We show that a multidimensional generalized Riemann integral defined by means of rectangular figures is already invariant with respect to lipeomorphic changes of coordinates.

The Stokes theorem for noncontinuously differentiable forms has been established by means of a coordinate free Riemann-type integral, which integrates the divergence of any differentiable vector field over bounded sets of finite perimeter. We show that the pointwise products of integrable and Lipschitz functions are integrable, and interpret the integrable functions as distributions.

Using functions of bounded variation, we define a Volterra type derivative of the linear functional associated with a Lebesgue integrable function and show that it is equal to this function almost everywhere.

Using ideas of McShane ([4, Example 3]), a detailed development of the Riemann integral in a locally compact Hausdorff space X was presented in [1]. There the Riemann integral is derived from a finitely additive volume nu defined on a suitable semiring of subsets of X. Vis-a-vis the Riesz representation theorem ([8, Theorem 2.14]), the integral generates a Riesz measure nu in X, whose relationship to the volume nu was carefully investigated in [1, Section 7]. In the present paper, we use the same setting as in [1] but produce the measure directly without introducing the Riemann integral. Specifically, we define an outer measure by means of gages and introduce a very intuitive concept of gage measurability that is different from the usual Caratheodory definition. We prove that if the outer measure is sigma-finite, the resulting measure space is identical to that defined by means of the Caratheodory technique, and consequently to that of [1, Section 7]. If the outer measure is not sigma-finite, we investigate the gage measurability of Caratheodory measurable sets that are sigma-finite. Somewhat surprisingly, it turns out that this depends on the axioms of set theory.

We present a Riemann type definition of a coordinate free integral for which a general divergence theorem holds. The definition is particularly simple in dimension one.

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.

We use a descriptive definition of a coordinate free integral to obtain a divergence theorem for vector fields which are discontinuous on a set of positive measure.

Proceedings of the London Mathematical SocietyVolume s3-58, Issue 3 p. 417-438 Articles Small Spaces Washek F. Pfeffer, Washek F. Pfeffer Department of Mathematics University of California, Davis California, 95616 U.S.A.Search for more papers by this authorKarel Prikry, Karel Prikry Department of Mathematics University of Minnesota, Minneapolis Minnesota, 55455 U.S.A.Search for more papers by this author Washek F. Pfeffer, Washek F. Pfeffer Department of Mathematics University of California, Davis California, 95616 U.S.A.Search for more papers by this authorKarel Prikry, Karel Prikry Department of Mathematics University of Minnesota, Minneapolis Minnesota, 55455 U.S.A.Search for more papers by this author First published: May 1989 https://doi.org/10.1112/plms/s3-58.3.417Citations: 9AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes3-58, Issue3May 1989Pages 417-438 RelatedInformation

We show that the generalized Riemann integral can be defined by means of gage functions which are upper semicontinuous when restricted to a suitable subset whose complement has measure zero.