viii Contents
§3.2. The Lebesgue Integral of Bounded Functions 48
§3.3. The Bounded Convergence Theorem 56
Chapter 4. The Integral of Unbounded Functions 63
§4.1. Non-negative Functions 63
§4.2. Convergence Theorems 67
§4.3. Other Measures 72
§4.4. General Measurable Functions 77
Chapter 5. The Hilbert Space L2 83
§5.1. Square Integrable Functions 83
§5.2. Convergence in L2 89
§5.3. Hilbert Space 95
§5.4. Fourier Series 99
§5.5. Complex Hilbert Space 104
Chapter 6. Classical Fourier Series 111
§6.1. Real Fourier Series 111
§6.2. Integrable Complex-Valued Functions 119
§6.3. The Complex Hilbert Space LC
2[−π,
π] 122
§6.4. The Hilbert Space LC
2[T]
125
Chapter 7. Two Ergodic Transformations 129
§7.1. Measure Preserving Transformations 130
§7.2. Ergodicity 134
§7.3. The Birkhoff Ergodic Theorem 137
Appendix A. Background and Foundations 141
§A.1. The Completeness of R 141
§A.2. Functions and Sequences 143
§A.3. Limits 145
§A.4. Complex Limits 148
§A.5. Set Theory and Countability 151
§A.6. Open and Closed Sets 156
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