This book aims at restructuring some fundamentals in measure and integration theory and thus to free the theory from notorious drawbacks. It centers around the ubiquitous task to produce appropriate contents and measures from more primitive data like elementary contents and elementary integrals. It develops the new approach started around 1970 by Topsoe and others into a systematic theory. The theory is much more powerful than the traditional means and has striking implications all over measure theory and beyond. Thus it extends the Riesz representation theorem in terms of Radon measures from locally compact to arbitrary Hausdorff topological spaces.
Author(s): Heinz König
Edition: 1
Publisher: Springer, Berlin
Year: 1997
Language: English
Pages: 282
Preface......Page 6
Contents......Page 8
1. Set Systems......Page 21
2. Set Functions......Page 30
3. Some Classical Extension Theorems for Set Functions......Page 42
4. The Outer Extension Theory: Concepts and Instruments......Page 53
5. The Outer Extension Theory: The Main Theorem......Page 65
6. The Inner Extension Theory......Page 73
7. Complements to the Extension Theories......Page 84
8. Baire Measures......Page 99
9. Radon Measures......Page 107
10. The Choquet Capacitability Theorem......Page 118
11. The Horizontal Integral......Page 129
12. The Vertical Integral......Page 141
13. The Conventional Integral......Page 148
14. Elementary Integrals on Lattice Cones......Page 163
15. The Continuous Daniell-Stone Theorem......Page 174
16. The Riesz Theorem......Page 185
17. The Non-continuous Daniell-Stone Theorem......Page 191
18. Transplantation of Contents......Page 199
19. Transplantation of Measures......Page 210
20. The Traditional Product Formations......Page 221
21. The Product Formations Based on Inner Regularity......Page 230
22. The Fubini-Tonelli and Fubini Theorems......Page 242
23. The Jordan and Hahn Decomposition Theorems......Page 251
24. The Lebesgue Decomposition and Radon-Nikod´ym Theorems......Page 262
Bibliography......Page 269
List of Symbols......Page 275
Index......Page 277
