Measure, Integration & Real Analysis, 2021a

Contents
ix
5 Product Measures 116
5A Products of Measure Spaces 117
Products of σ-Algebras 117
Monotone Class Theorem 120
Products of Measures 123
Exercises 5A 128
5B Iterated Integrals 129
Tonelli’s Theorem 129
Fubini’s Theorem 131
Area Under Graph 133
Exercises 5B 135
5C Lebesgue Integration on R n 136
6 Banach Spaces 146
Borel Subsets of R n 136
Lebesgue Measure on R n 139
Volume of Unit Ball in R n 140
Equality of Mixed Partial Derivatives Via Fubini’s Theorem 142
Exercises 5C 144
6A Metric Spaces 147
Open Sets, Closed Sets, and Continuity 147
Cauchy Sequences and Completeness 151
Exercises 6A 153
6B Vector Spaces 155
Integration of Complex-Valued Functions 155
Vector Spaces and Subspaces 159
Exercises 6B 162
6C Normed Vector Spaces 163
Norms and Complete Norms 163
Bounded Linear Maps 167
Exercises 6C 170
6D Linear Functionals 172
Bounded Linear Functionals 172
Discontinuous Linear Functionals 174
Hahn–Banach Theorem 177
Exercises 6D 181
Measure, Integration & Real Analysis, by Sheldon Axler