An introduction to nonstandard real analysis

Cover......Page 1
Title Page......Page 5
Contents......Page 9
Preface......Page 11
I Infinitesimals and The Calculus......Page 15
I.1 The Hyperreal Number System as an Ultrapower......Page 16
I.2 *-Transforms of Relations......Page 22
I.3 Simple Languages for Relational Systems......Page 25
I.4 Interpretation of Simple Sentences......Page 29
I.5 The Transfer Principle for Simple Sentences......Page 33
I.6 Infinite Numbers, lnfinitesimals, and the Standard Part Map......Page 38
I.7 The Hyperintegers......Page 43
I.8 Sequences and Series......Page 46
I.9 Topology on the Reals......Page 53
I.10 Limits and Continuity......Page 58
I.11 Differentiation......Page 65
I.12 Riemann Integration......Page 70
I.13 Sequences of Functions......Page 74
I.14 Two Applications to Differential Equations......Page 77
I.15 Proof of the Transfer Principle......Page 81
II Nonstandard Analysis on Superstructures......Page 84
II.1 Superstructures......Page 85
II.2 Languages and Interpretation for Superstructures......Page 88
II.3 Monomorphisms between Superstructures: The Transfer Principle......Page 92
II.4 The Ultrapower Construction for Superstructures......Page 97
II.5 Hyperfinite Sets, Enlargements, and Concurrent Relations......Page 102
II.6 Internal and External Entities; Comprehensiveness......Page 108
II.7 The Permanence Principle......Page 114
II.8 κ-Saturated Superstructures......Page 118
III Nonstandard Theory of Topological Spaces......Page 123
III.1 Basic Definitions and Results......Page 124
III.2 Compactness......Page 134
III.3 Metric Spaces......Page 137
III.4 Normed Vector Spaces and Banach Spaces......Page 146
III.5 Inner-Product Spaces and Hilbert Space......Page 159
III.6 Nonstandard Hulls of Metric Spaces......Page 168
III.7 Compactifications......Page 170
III.8 Function Spaces......Page 174
IV Nonstandard Integration Theory......Page 178
IV.1 Standardizations of Internal Integration Structures......Page 179
IV.2 Measure Theory for Complete Integration Structures......Page 189
IV.3 Integration on ℝⁿ; the Riesz Representation Theorem......Page 203
IV.4 Basic Convergence Theorems......Page 209
IV.5 The Fubini Theorem......Page 214
IV.6 Applications to Stochastic Processes......Page 219
Appendix: Ultrafilters......Page 233
References......Page 236
List of Symbols......Page 239
Index......Page 241