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1．Measures 1

1．Algebras and sigma-algebras 1

2．Measures 8

3．Outer measures 14

4．Lebesgue measure 26

5．Completeness and regularity 35

6．Dynkin classes 44

2．Functions and Integrals 48

1．Measurable functions 48

2．Properties that hold almost everywhere 58

3．The integral 61

4．Limit theorems 70

5．The Riemann integral 75

6．Measurable functions again,complex-valued functions,and image measures 79

3．Convergence 85

1．Modes of convergence 85

2．Normed spaces 90

3．Definition of ？p and Lp 98

4．Properties of ？p and Lp 106

5．Dual spaces 113

4．Signed and Complex Measures 121

1．Signed and complex measures 121

2．Absolute continuity 131

3．Singularity 140

4．Functions of bounded variation 143

5．The duals of the Lp spaces 149

5．Product Measures 154

1．Constructions 154

2．Fubini's theorem 158

3．Applications 162

6．Differentiation 167

1．Change of variable in Rd 167

2．Differentiation of measures 177

3．Differentiation of functions 184

7．Measures on Locally Compact Spaces 196

1．Locally compact spaces 196

2．The Riesz representation theorem 205

3．Signed and complex measures;duality 217

4．Additional properties of regular measures 226

5．The μ＊-measurable sets and the dual of L1 232

6．Products of locally compact spaces 240

8．Polish Spaces and Analytic Sets 251

1．Polish spaces 251

2．Analytic sets 261

3．The separation theorem and its consequences 272

4．The measurability of analytic sets 278

5．Cross sections 284

6．Standard,analytic,Lusin,and Souslin spaces 288

9．Haar Measure 297

1．Topological groups 297

2．The existence and uniqueness of Haar measure 303

3．Properties of Haar measure 312

4．The algebras L1(G)and M(G) 317

Appendices 328

A．Notation and set theory 328

B．Algebra 334

C．Calculus and topology in Rd 339

D．Topological spaces and metric spaces 342

E．The Bochner integral 350

Bibliography 361

Index of notation 367

Index 369