A First Course in Integration

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CoverA FIRST COURSE IN INTEGRATIONCOPYRIGHT © 1966 BY HOLT, RINEHART AND WINSTON, INC LCCN 20550-0116PrefaceContents1 Step Functions and Null Sets 1.0 THE REAL LINE 1.1 STEP FUNCTIONS AND THEIR INTEGRALS 1.2 BASIC PROPERTIES OF THE INTEGRAL 1.3 NULL SETS 1.4 EXAMPLES 1.5 CONVERGENCE ALMOST EVERYWHERE 1.6 EXERCISES A. Step Functions and Their Integral B. Null Sets C. Convergence D. Piecewise Linear Functions SUMMARY2 The Lebesgue Integral 2.0 THE RIEMANN INTEGRAL FOR CONTINUOUS FUNCTIONS 2.1 LEBESGUE INTEGRABLE FUNCTIONS 2.2 BASIC PROPERTIES OF THE LEBESGUE INTEGRAL 2.3 CONVERGENCE THEOREMS 2.4 APPLICATIONS 2.5 THE DANIELL INTEGRAL 2.6 RIEMANN INTEGRABLE FUNCTIONS 2.7 EXERCISES A. Basic Properties of Lebesgue Integrable Functions B. Problems on Limits (Abstract) C. Problems on Limits (Concrete) D. Product of Functions E. Improper Riemann Integration G. The Daniell Integral SUMMARY BEPPO LEVI'S THEOREM MONOTONE CONVERGENCE THEOREM LEBESGUE'S DOMINATED CONVERGENCE THEOREM FATOU'S LEMMA C0MPAJUs0N THEOREM3 Measurability 3.1 MEASURABLE FUNCTIONS 3.2 GENERALIZED STEP FUNCTIONS 3.3 MEASURABLE SETS EGOROFF'S THEOREM STEINHAUS THEOREM 3.4 AXIOMATIC MEASURE THEORY 3.5 OUTER MEASURE 3.6 EXERCISES A. Measurable Functions and Measurable Sets B. Relation between Measurability and Integrability C. Baire Classes of Functions of k Variables D. Composition of Measurable Functions with Functions in Gn E. Lusin's Theorem F. Outer and Inner Measure G. The Cauchy Functional Equation H. Vitali's Covering Theorem SUMMARY4 Integration of Functions of Several Variables 4.1 DEFINITIONS AND FUBINI'S THEOREM 4.2 THE LEBESGUE INTEGRAL ON R^n 4.3 LINEAR CHANGE OF VARIABLES 4.4 NONLINEAR CHANGE OF VARIABLES 4.5 EXERCISES A. Product Integration B. The Lebesgue Integral for R^n C. Linear Change of Variables D. Nonlinear Change of Variables E. Integration on the Unimodular Group SUMMARY5 L^2 and L^p Spaces 5.1 THE SPACE L^2 5.2 METRIC PROPERTIES OF L^2 5.3 LINEAR FUNCTIONALS ON L^2 5.4 L^p SPACES 5.5 INTEGRATION OF COMPLEXVALUED FUNCTIONS 5.6 THE FOURIER TRANSFORMATION 5.7 EXERCISES A. Strong and Weak Convergence B. Orthonormal Systems C. The Space l^2 D. Continuous Quadratic Functionals on L^2 E. Continuous Linear Functionals on L^p F. Complex L^2 and Fourier Transforms G. Fourier Transformations for Functions of Several Variables SUMMARY6 The Differentiation of Functions of Locally Bounded Variation 6.1 FUNCTIONS OF LOCALLY BOUNDED VARIATION 6.2 DECOMPOSITION OF FUNCTIONS OF LOCALLY BOUNDED VARIATION 6.3 PREPARATIONS FOR THE DIFFERENTIATION THEOREM 6.4 DIFFERENTIATION OF MONOTONENON DECREASING FUNCTIONS 6.5 DIFFERENTIATION OF INFINITE SERIES 6.6 EXERCISES A. Functions of Locally Bounded Variation B. Convergence Theorems C. Functions of Bounded Variation D. Differentiation E. The Density Theorem and the Theorem on Derivates 6.7 INTEGRATION AS INVERSE OF DIFFERENTIATION SUMMARY7 Absolutely Continuous Functions 7.1 INTEGRATION OF THE DERIVATIVE OF A FUNCTION IN V 7.2 THE PRIMITIVE OF A LOCALLY INTEGRABLE FUNCTION 7.3 THE FUNDAMENTAL THEOREM FOR ABSOLUTELY CONTINUOUS FUNCTIONS 7.4 ANOTHER CHARACTERIZATION OF ABSOLUTELY CONTINUOUS FUNCTIONS 7.5 APPLICATIONS 7.6 EXERCISES A. The Indefinite Integral B. Examples of Absolutely Continuous Functions C. Convergence of Absolutely Continuous Functions D. Mapping Properties of Absolutely Continuous Functions E. Composition of Absolutely Continuous Functions F. Discussion of Theorem 7.5.118 Stieltj es Integrals 8.1 DEFINITION OF THE STIELTJES INTEGRAL 8.2 PROPERTIES OF STIELTJES INTEGRALS 8.3 CONTINUOUS LINEAR FUNCTIONALS ON C 8.4 OPERATIONS ON FUNCTIONALS 8.5 THE RIESZ REPRESENTATION THEOREM 8.6 EXERCISES A. Examples of Stieltjes Integrals B. Stieltjes Integrals for Piecewise Linear Functions C. Baire Functions D. Limit Operations E. Operations on Continuous Linear Functionals on C F. A Theory of Integration for Signed Stieltjes Integrals G. Abstract Signed Integrals SUMMARY RIEsz REPRESENTATION THEOREM9 The Radon-Nikodym Theorem 9.1 ABSOLUTE CONTINUITY 9.2 THE RADON-NIKODYM THEOREM 9.3 THE ABSTRACT RADON-NIKODYM THEOREM 9.4 APPLICATIONS 9.5 EXERCISES A. Absolute Continuity B. Relations between Radon-Nikodym Derivatives C. Lebesgue Decomposition of Measures D. Absolute Continuity for Signed Measures SUMMARY10 Applications to the Theory of Fourier Series 10.1 TRIGONOMETRIC SERIES AND FOURIER SERIES 10.2 CONVERGENCE TESTS 10.3 SUMMATION BY ARITHMETIC MEANS 10.4 FOURIER SERIES FOR FUNCTIONSIN L2(O, 2 \pi) 10.5 EXERCISES A. Examples of Fourier Series B. Trigonometric Series with Monotone-Decreasing Coefficients C. The Fourier Coefficients of Some Classes of Functions D. Summation by the Method of Arithmetic Means E. Fourier Series for Functions in L^2 SUMMARYBibliographical Comments and Remarks COMMENTS ON THE REFERENCES REFERENCES REMARKS ON THE LITERATURE LITERATUREIndexes THEOREMS REFERRED TO BY NAME SYMBOLS SUBJECT INDEX