GEOMETRIC INTEGRATION THEORY

By J. GANIY

CONTENTS BASICS SMOOTH FUNCTIONS MEASURES LEBESGUE MEASURE INTEGRATION MEASURABLE FUNCTIONS THE INTEGRAL LEBESGUE SPACES PRODUCT MEASURES AND THE FUBINI–TONELLI THEOREM THE EXTERIOR ALGEBRA THE HAUSDOR?DISTANCE AND STEINER SYMMETRIZATION BOREL AND SUSLIN SETS CARATH´EODORY’S CONSTRUCTION AND LOWER-DIMENSIONAL MEA- SURES THE BASIC DE?NITION HAUSDOR?MEASURE AND SPHERICAL MEASURE A MEASURE BASED ON PARALLELEPIPEDS PROJECTIONS AND CONVEXITY OTHER GEOMETRIC MEASURES SUMMARY THE DENSITIES OF A MEASURE A ONE-DIMENSIONAL EXAMPLE CARATH´EODORY’S CONSTRUCTION AND MAPPINGS THE CONCEPT OF HAUSDOR?DIMENSION SOME CANTOR SET EXAMPLES BASIC EXAMPLES SOME GENERALIZED CANTOR SETS CANTOR SETS IN HIGHER DIMENSIONS INVARIANT MEASURES AND THE CONSTRUCTION OF HAAR MEASURE THE FUNDAMENTAL THEOREM HAAR MEASURE FOR THE ORTHOGONAL GROUP AND THE GRASSMANIAN REMARKS ON THE MANIFOLD STRUCTURE OF G(N, M) COVERING THEOREMS AND THE DI?ERENTIATION OF INTEGRALS WIENER’S COVERING LEMMA AND ITS VARIANTS THE BESICOVITCH COVERING THEOREM DECOMPOSITION AND DI?ERENTIATION OF MEASURES MAXIMAL FUNCTIONS REDUX ANALYTICAL TOOLS: THE AREA FORMULA, THE COAREA FORMULA, AND POINCAR´E INEQUALITIES THE AREA FORMULA LINEAR MAPS C1 FUNCTIONS RADEMACHER’S THEOREM THE COAREA FORMULA MEASURE THEORY OF LIPSCHITZ MAPS PROOF OF THE COAREA FORMULA THE AREA AND COAREA FORMULAS FOR C1 SUBMANIFOLDS RECTI?ABLE SETS POINCAR´E INEQUALITIES THE CALCULUS OF DI?ERENTIAL FORMS AND STOKES’S THEOREM DI?ERENTIAL FORMS AND EXTERIOR DI?ERENTIATION STOKES’S THEOREM INTRODUCTION TO CURRENTS A FEW WORDS ABOUT DISTRIBUTIONS THE DE?NITION OF A CURRENT CONSTRUCTIONS USING CURRENTS AND THE CONSTANCY THEOREM FURTHER CONSTRUCTIONS WITH CURRENTS PRODUCTS OF CURRENTS THE PUSH-FORWARD THE HOMOTOPY FORMULA APPLICATIONS OF THE HOMOTOPY FORMULA RECTI?ABLE CURRENTS WITH INTEGER MULTIPLICITY SLICING THE DEFORMATION THEOREM PROOF OF THE UNSCALED DEFORMATION THEOREM APPLICATIONS OF THE DEFORMATION THEOREM CURRENTS AND THE CALCULUS OF VARIATIONS PROOF OF THE COMPACTNESS THEOREM INTEGER-MULTIPLICITY 0-CURRENTS A RECTI?ABILITY CRITERION FOR CURRENTS MBV FUNCTIONS THE SLICING LEMMA THE DENSITY LEMMA COMPLETION OF THE PROOF OF THE COMPACTNESS THEOREM THE FLAT METRIC EXISTENCE OF CURRENTS MINIMIZING VARIATIONAL INTEGRALS MINIMIZING MASS OTHER INTEGRANDS AND INTEGRALS DENSITY ESTIMATES FOR MINIMIZING CURRENTS REGULARITY OF MASS-MINIMIZING CURRENTS PRELIMINARIES THE HEIGHT BOUND AND LIPSCHITZ APPROXIMATION CURRENTS DE?NED BY INTEGRATING OVER GRAPHS ESTIMATES FOR HARMONIC FUNCTIONS THE MAIN ESTIMATE THE REGULARITY THEOREM EPILOGUE BIBLIOGRAPHY