The general objective of this treatise is to give a systematic presentaยญ tion of some of the topological and measure-theoretical foundations of the theory of real-valued functions of several real variables, with particular emphasis upon a line of thought initiated by BANACH, GEOCZE, LEBESGUE, TONELLI, and VITALI. To indicate a basic feature in this line of thought, let us consider a real-valued continuous function I(u) of the single real variable tt. Such a function may be thought of as defining a continuous translormation T under which x = 1 (u) is the image of u. About thirty years ago, BANACH and VITALI observed that the fundamental concepts of bounded variation, absolute continuity, and derivative admit of fruitful geometrical descriptions in terms of the transformation T: x = 1 (u) associated with the function 1 (u). They further noticed that these geometrical descriptions remain meaningful for a continuous transformation T in Euclidean n-space Rff, where T is given by a system of equations of the form 1-/(1 ff) X-I U, . . . ,tt ,. ", and n is an arbitrary positive integer. Accordingly, these geometrical descriptions can be used to define, for continuous transformations in Euclidean n-space Rff, n-dimensional concepts 01 bounded variation and absolute continuity, and to introduce a generalized Jacobian without reference to partial derivatives. These ideas were further developed, generalized, and modified by many mathematicians, and significant applications were made in Calculus of Variations and related fields along the lines initiated by GEOCZE, LEBESGUE, and TONELLI
CONTENT
I. Background in Topology -- ยง I.1. Survey of general topology -- ยง I.2. Survey of Euclidean spaces -- ยง I.3. Survey of Abelian groups -- ยง I.4. Mayer complexes -- ยง I.5. Formal complexes -- ยง I.6. General cohomology theory -- ยง I.7. Cohomology groups in Euclidean spaces -- II. Topological study of continuous transformations in Rn -- ยง II.1. Orientation in Rn -- ยง II.2. The topological index -- ยง II.3. Multiplicity functions and index functions -- III. Background in Analysis -- ยง III.1. Survey of functions of real variables -- ยง III.2. Functions of open intervals in Rn -- IV. Bounded variation and absolute continuity in Rn -- ยง IV.1. Measurability questions -- ยง IV.2. Bounded variation and absolute continuity with respect to a base-function -- ยง IV.3. Bounded variation and absolute continuity with respect to a multiplicity function -- ยง IV.4. Essential bounded variation and absolute continuity -- ยง IV.5. Bounded variation and absolute continuity in the Banach sense -- V. Differentiable transformations in Rn -- ยง V.1. Continuous transformations in R1 -- ยง V.2. Local approximations in Rn -- ยง V.3. Special classes of differentiable transformations in Rn -- VI. Continuous transformations in R2 -- ยง VI.1. The topological index in R2 -- ยง VI.2. Special features of continuous transformations in R2 -- ยง VI.3. Special classes of differentiable transformations in R2
