Washek F. Pfeffer

The multidimensional fundamental theorem of calculus

On compact oriented differentiable manifolds, we define a well behaved Riemann type integral which coincides with the Lebesgue integral on nonnegative functions, and such that the exterior derivative of a differentiable (not necessarily continuously) exterior form is always integrable and the Stokes formula holds.

The Gauss-Green theorem

Abstract In the m -dimensional Euclidean space, we establish the Gauss-Green theorem for any bounded set of bounded variation, and any bounded vector field continuous outside a set of ( m − 1)-dimensional Hausdorff measure zero and almost differentiable outside a set of σ-finite ( m − 1)-dimensional Hausdorff measure.

Variations of additive functions

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 Wards theorem, a necessary and sufficient condition for derivability, and full descriptive definitions of certain conditionally convergent integrals.

The divergence theorem and sets of finite perimeter

DYADIC FIGURES Preliminaries The setting Topology Measures Hausdorff measures Differentiable and Lipschitz maps Divergence Theorem for Dyadic Figures Differentiable vector fields Dyadic partitions Admissible maps Convergence of dyadic figures Removable Singularities Distributions Differential equations Holomorphic functions Harmonic functions The minimal surface equation Injective limits SETS OF FINITE PERIMETER Perimeter Measure-theoretic concepts Essential boundary Vitalis covering theorem Density Definition of perimeter Line sections BV Functions Variation Mollification Vector valued measures Weak convergence Properties of BV functions Approximation theorem Coarea theorem Bounded convex domains Inequalities Locally BV Sets Dimension one Besicovitchs covering theorem The reduced boundary Blow-up Perimeter and variation Properties of BV sets Approximating by figures THE DIVERGENCE THEOREM Bounded Vector Fields Approximating from inside Relative derivatives The critical interior The divergence theorem Lipschitz domains Unbounded Vector Fields Minkowski contents Controlled vector fields Integration by parts Mean Divergence The derivative The critical variation Charges Continuous vector fields Localized topology Locally convex spaces Duality The space BVc(OMEGA) Streams The Divergence Equation Background Solutions in Lp(OMEGA Rn) Continuous solutions Bibliography List of Symbols Index

The Gauss–Green theorem and removable sets for PDEs in divergence form

Abstract 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.

Integration by parts for the generalized Riemann-Stieltjes integral

Under the standard assumptions, the formula for integration by parts is obtained by a straightforward calculation.

Some undecidability results concerning Radon measures

We show that in metalindelof spaces certain questions about Radon measures cannot be decided within the Zermelo-Fraenkel set theory, including the axiom of choice. 0. Introduction. All spaces in this paper will be Hausdorff. Let X be a space. By g and e we shall denote, respectively, the families of all open and compact subsets of X. The Borel a-algebra in X, i.e., the smallest a-algebra in X containing g, will be denoted by 613. The elements of I are called Borel sets. A Borel measure in X is a measure It on 133 such that each x E X has a neighborhood U E 1i3 with ,u(U) < +x0. A Borel measure It in X is called (i) Radon if (B) = sup{ 1(C): C E (C, C c B) for each B E I; (ii) regular if 1(B) inf{ t(G): G E 9, B c G) for each B E 03 . A Radon space is a space X in which each finite Borel measure is Radon. The present paper will be divided into two parts, each dealing with one of the following questions. (a) Let X be a locally compact, hereditarily metalindelof space with the countable chain condition. Is X a Radon space? (,B) Let X be a metalindelof space, and let It be a a-finite Radon measure in X. Is IL a regular measure? We shall show that neither question can be decided within the usual axioms of set theory. The proofs are based on an alternate application of the continuum hypothesis, Martins axiom and Ostaszewskis axiom 4. In passing, we shall also answer a question of F. D. Tall (see [T, last paragraph in ?6]). Namely, using the continuum hypothesis and 4, we shall construct a locally compact, hereditarily metalindelof space X with the countable chain condition which is not weakly 0-refinable. Received by the editors July 6, 1978. AMS (MOS) subject classifications (1970). Primary 54D20, 54G20, 54H99; Secondary 28A30, 28A35.

A note on the generalized Riemann integral

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. By introducing S-fine partitions for a positive function 6 (see below), Henstock and Kurzweil obtained a strikingly simple Riemannian definition of the DenjoyPeron integral (cf. Definition 1 and [S, Chapter VIII]). In their definition, the function 6 is completely arbitrary, and it is not clear how complicated it need be (a question of P. S. Bullen see [Q]). The purpose of this note is to establish that 6 can be always selected so that it is upper semicontinuous when restricted to a suitable subset whose complement has measure zero (cf. [P2, Lemma 3]). The proof is quite simple: we show first that such a 6 can be chosen if the integrand is Lebesgue integrable, and then we follow the constructive Denjoy definition, observing that the upper semicontinuity property of 6 is preserved at the inductive step. We also show that for a bounded Lebesgue integrable function, a gage 6 can be selected so that it is upper semicontinuous everywhere (cf. [FM, Example 1]). The author is obliged to J. Foran for pointing out a serious error in the preprint of this paper. By R and R+ we denote the set of all real and all positive real numbers, respectively. Unless stated otherwise, all functions in this paper are real-valued. When no confusion is possible, we denote by the same symbol a function on a set E, as well as its restrictions to various subsets of E. An interval is a compact nondegenerate subinterval of R. A collection of intervals whose interiors are disjoint is called a nonoverlapping collection. If E c R, then cl(E), int(E), d(E), and JEl denote, respectively, the closure, interior, diameter, and outer Lebesgue measure of E. A function 6 on an interval A is called nearly upper semicontinuous if there is a set H C A such that IA HI = 0 and 6 [ H is upper semicontinuous. A subpartition of an interval A is a collection P = {(A1, xi), .. . , (Ap, xp)} where A1,... , AP are nonoverlapping subintervals of A, and xi E Ai, i = 1,... , p. If, in addition, Up=1 Ai = A, we say that P is a partition of A. Given a 6: A -R+, we say that a subpartition P is 6-fine whenever d(Ai) < 6(xi) for i = 1,... ,p. An easy compactness argument shows that a 8-fine partition of an interval A exists for each 8: A -R+. Received by the editors June 11, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 26A39. (?)1988 American Mathematical Society 0002-9939/88

Luzin’s theorem for charges

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A Volterra type derivative of the Lebesgue integral

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