Contents Preface xi Chapter 1. Setting the Stage 1 §1.1. Riemann–Stieltjes integrals 1 §1.2. Metric spaces redux 12 §1.3. Normed spaces 21 §1.4. Locally compact spaces 29 §1.5. Linear functionals 37 Chapter 2. Elements of Measure Theory 41 §2.1. Positive linear functionals on C(X) 41 §2.2. The Radon measure space 42 §2.3. Measurable functions 51 §2.4. Integration with respect to a measure 56 §2.5. Convergence theorems 71 §2.6. Signed measures 78 §2.7. Lp-spaces 84 Chapter 3. A Hilbert Space Interlude 93 §3.1. Introduction to Hilbert space 93 §3.2. Orthogonality 98 §3.3. The Riesz Representation Theorem 103 Chapter 4. A Return to Measure Theory 107 §4.1. The Lebesgue–Radon–Nikodym Theorem 107 vii
Previous Page Next Page