CONTENTS 5

12 Series and Transforms 27712.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27712.2 Criteria for Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 27912.3 Integrating and Differentiating Fourier Series . . . . . . . . . . . . . . . . 28212.4 Ways of Approximating Functions . . . . . . . . . . . . . . . . . . . . . 28512.5 Uniform Approximation with Trig. Polynomials . . . . . . . . . . . . . . 28512.6 Mean Square Approximation . . . . . . . . . . . . . . . . . . . . . . . . 28712.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28912.8 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29212.9 The Inversion of Laplace Transforms . . . . . . . . . . . . . . . . . . . . 29512.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

13 The Generalized Riemann Integral 30113.1 Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . 30113.2 Monotone Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . 30813.3 Computing Generalized Integrals . . . . . . . . . . . . . . . . . . . . . . 31113.4 Integrals of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 31413.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

14 The Lebesgue Integral 32114.1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32114.2 Dynkin’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32414.3 The Lebesgue Stieltjes Measures and Borel Sets . . . . . . . . . . . . . . 32514.4 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32814.5 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33014.6 Riemann Integrals for Decreasing Functions . . . . . . . . . . . . . . . . 33414.7 Lebesgue Integrals of Nonnegative Functions . . . . . . . . . . . . . . . . 33414.8 Nonnegative Simple Functions . . . . . . . . . . . . . . . . . . . . . . . 33514.9 The Monotone Convergence Theorem . . . . . . . . . . . . . . . . . . . 33714.10 The Integral’s Righteous Algebraic Desires . . . . . . . . . . . . . . . . . 33814.11 Integrals of Real Valued Functions . . . . . . . . . . . . . . . . . . . . . 33814.12 The Vitali Covering Theorems . . . . . . . . . . . . . . . . . . . . . . . 34114.13 Differentiation of Increasing Functions . . . . . . . . . . . . . . . . . . . 34614.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

15 Construction of Real Numbers 355

A Classification of Real Numbers 359A.1 Algebraic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359A.2 The Symmetric Polynomial Theorem . . . . . . . . . . . . . . . . . . . . 360A.3 Transcendental Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 364

B Integration on Rough Paths∗ 373B.1 Finite p Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374B.2 Piecewise Linear Approximation . . . . . . . . . . . . . . . . . . . . . . 377B.3 The Young Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379Copyright © 2018, You are welcome to use this, including copying it for use in classes

or referring to it on line but not to publish it for money. I do constantly upgrade it when Ifind things which could be improved. This happens often.