ELEMENTS OF MATHEMATICS ANALYSIS

By T. DAMID

CONTENTS MEASURE SYSTEMS OF SETS VOLUME PROPERTIES OF MEASURE EXTENSION OF MEASURE PROPERTIES OF THE CARATHÉODORY EXTENSION ?PROPERTIES OF THE BOREL HULL OF A SYSTEM OF SETS THE LEBESGUE MEASURE DE?NITION AND BASIC PROPERTIES OF THE LEBESGUE MEASURE REGULARITY OF THE LEBESGUE MEASURE PRESERVATION OF MEASURABILITY UNDER SMOOTH MAPS INVARIANCE OF THE LEBESGUE MEASURE UNDER RIGID MOTIONS BEHAVIOR OF THE LEBESGUE MEASURE UNDER LINEAR MAPS ?HAUSDORFF MEASURES ?THE VITALI THEOREM ?THE BRUNN–MINKOWSKI INEQUALITY MEASURABLE FUNCTIONS DE?NITION AND BASIC PROPERTIES OF MEASURABLE FUNCTIONS SIMPLE FUNCTIONS THE APPROXIMATION THEOREM CONVERGENCE IN MEASURE AND CONVERGENCE ALMOST EVERYWHERE ?APPROXIMATION OF MEASURABLE FUNCTIONS BY CONTINUOUS FUNCTIONS LUZIN’S THEOREM THE INTEGRAL DE?NITION OF THE INTEGRAL PROPERTIES OF THE INTEGRAL OF NON-NEGATIVE FUNCTIONS PROPERTIES OF THE INTEGRAL RELATED TO THE “ALMOST EVERYWHERE” NOTION PROPERTIES OF THE INTEGRAL OF SUMMABLE FUNCTIONS THE INTEGRAL AS A SET FUNCTION THE LEBESGUE INTEGRAL OF A FUNCTION OF ONE VARIABLE THE MULTIPLE LEBESGUE INTEGRAL INTERCHANGE OF LIMITS AND INTEGRATION ?THE MAXIMAL FUNCTION AND DIFFERENTIATION OF THE INTEGRAL WITH RESPECT TO A SET 4 10 ?THE LEBESGUE–STIELTJES MEASURE AND INTEGRAL 4 11 ?FUNCTIONS OF BOUNDED VARIATION THE PRODUCT MEASURE DE?NITION OF THE PRODUCT MEASURE THE COMPUTATION OF THE MEASURE OF A SET VIA THE MEASURES OF ITS CROSS SECTIONS THE INTEGRAL AS THE MEASURE OF THE SUBGRAPH DOUBLE AND ITERATED INTEGRALS LEBESGUE MEASURE AS A PRODUCT MEASURE ?AN ALTERNATIVE APPROACH TO THE DE?NITION OF THE PRODUCT MEASURE AND THE INTEGRAL ?IN?NITE PRODUCTS OF MEASURES CHANGE OF VARIABLES IN AN INTEGRAL INTEGRATION OVER A WEIGHTED IMAGE OF A MEASURE CHANGE OF VARIABLE IN A MULTIPLE INTEGRAL INTEGRAL REPRESENTATION OF ADDITIVE FUNCTIONS ?DISTRIBUTION FUNCTIONS INDEPENDENT FUNCTIONS ?COMPUTATION OF MULTIPLE INTEGRALS BY INTEGRATING OVER THE SPHERE ?SOME GEOMETRIC APPLICATIONS ?SOME GEOMETRIC APPLICATIONS (CONTINUED) INTEGRALS DEPENDENT ON A PARAMETER BASIC THEOREMS ?THE GAMMA FUNCTION ?THE LAPLACE METHOD ?IMPROPER INTEGRALS DEPENDENT ON A PARAMETER EXISTENCE CONDITIONS AND BASIC PROPERTIES OF CONVOLUTION APPROXIMATE IDENTITIES SURFACE INTEGRALS AUXILIARY NOTIONS SURFACE AREA PROPERTIES OF THE SURFACE AREA OF A SMOOTH MANIFOLD INTEGRATION OVER A SMOOTH MANIFOLD INTEGRATION OF VECTOR FIELDS THE GAUSS–OSTROGRADSKI FORMULA ?HARMONIC FUNCTIONS ?AREA ON LIPSCHITZ MANIFOLDS APPROXIMATION AND CONVOLUTION IN THE SPACES L P THE SPACES L P ?APPROXIMATION IN THE SPACES L P ?CONVOLUTION AND APPROXIMATE IDENTITIES IN THE SPACES L P FOURIER SERIES AND THE FOURIER TRANSFORM 10 1 ORTHOGONAL SYSTEMS IN THE SPACE L 2(X,?) 10 2 ?EXAMPLES OF ORTHOGONAL SYSTEMS 10 3 TRIGONOMETRIC FOURIER SERIES 10 4 ?TRIGONOMETRIC FOURIER SERIES (CONTINUED) 10 5 THE FOURIER TRANSFORM 10 6 ?THE POISSON SUMMATION FORMULA CHARGES THE RADON–NIKODYM THEOREM 11 1 CHARGES; INTEGRATION WITH RESPECT TO A CHARGE 11 2 THE RADON–NIKODYM THEOREM 11 3 ?DIFFERENTIATION OF MEASURES 11 4 ?DIFFERENTIABILITY OF LIPSCHITZ FUNCTIONS INTEGRAL REPRESENTATION OF LINEAR FUNCTIONALS 12 1 ?ORDER CONTINUOUS FUNCTIONALS IN SPACES OF MEASURABLE FUNCTIONS 12 2 ?POSITIVE FUNCTIONALS IN SPACES OF CONTINUOUS FUNCTIONS 12 3 ?BOUNDED FUNCTIONALS APPENDICES 13 1 AN AXIOMATIC DE?NITION OF THE INTEGRAL OVER AN INTERVAL 13 2 EXTENSION OF CONTINUOUS FUNCTIONS 13 3 REGULAR MEASURES 13 4 CONVEXITY 13 5 SARD’S THEOREM 13 6 INTEGRATION OF VECTOR-VALUED FUNCTIONS 13 7 SMOOTH MAPS REFERENCES