哥德布拉特編著的《超實講義》是一部講述非標準分析的入門教程，是由作者數年教學講義發展並擴充而成。具備基本分析知識的高年級本科生，研究生以及自學人員都可以**讀懂。非標準分析理論不僅是研究無限大和無限小的強有力的理論，也是一種截然不同於標準數學概念和結構的方法，*是新的結構，目標和證明的源泉，推理原理的新起點。書中是從超實數繫統開始，從非標準的角度講述單變量積分，分析和拓撲，著重強調變換原理作為一個重要的數學工具的重要作用。數學宇宙的講述為全面研究非標準方法論提供了基礎保證。*後一章著眼於應用，將這些理論應用於loeb測度理論及其與lebesgue 的一些關繫，ramsey 定理，p-進數的非標準結構和冪級數，boolean 代數的stone 表示定理的非標準證明和hahn-banach 定理。《超實講義》的*大特點盡早引入內集，外集，超有限集，以及集理論擴展方法，較常規的建立在超結構基礎上，這樣的方式*加顯而易見。
讀者對像：數學專業的高年級本科生，研究生和科研人員。

I Foundations
1 What Are the Hyperreals?
1.1 Infinitely Small and Large
1.2 Historical Background
1.3 What Is a Real Number?
1.4 Historical References
2 Large Sets
2.1 Infinitesimals as Variable Quantities
2.2 Largeness
2.3 Filters
2.4 Examples of Filters
2.5 Facts About Filters
2.6 Zorn's Lemma
2.7 Exercises on Filters
3 Ultrapower Construction of the Hyperreals
3.1 The Ring of Real-Valued Sequences
3.2 Equivalence Modulo an Ultrafilter
3.3 Exercises on Almost-Everywhere Agreement
3.4 A Suggestive Logical Notation
3.5 Exercises on Statement Values
3.6 The Ultrapower
3.7 Including the Reals in the Hyperreals
3.8 Infinitesimals and Unlimited Numbers
3.9 Enlarging Sets
3.10 Exercises on Enlargement
3.11 Extending Functions
3.12 Exercises on Extensions
3.13 Partial Functions and Hypersequences
3.14 Enlarging Relations
3.15 Exercises on Enlarged Relations
3.16 Is the Hyperreal System Unique?
4 The Transfer Principle
4.1 Transforming Statements
4.2 Relational Structures
4.3 The Language of a Relational Structure
4.4 ,-Transforms
4.5 The Transfer Principle
4.6 Justifying Transfer
4.7 Extending Transfer
5 Hyperreals Great and Small
5.1 (Un)limited, Infinitesimal, and Appreciable Numbers
5.2 Arithmetic of Hyperreals
5.3 On the Use of "Finite" and "Infinite"
5.4 Halos, Galaxies, and Real Comparisons
5.5 Exercises on Halos and Galaxies
5.6 Shadows
5.7 Exercises on Infinite Closeness
5.8 Shadows and Completeness
5.9 Exercise on Dedekind Completeness
5.10 The Hypernaturals
5.11 Exercises on Hyperintegers and Primes
5.12 On the Existence of Infinitely Many Primes
II Basic Analysis
6 Convergence of Sequences and Series
6.1 Convergence
6.2 Monotone Convergence
6.3 Limits
6.4 Boundedness and Divergence
6.5 Cauchy Sequences
6.6 C, lustp.r Pnints
6.7 Exercises on Limits and Cluster Points
6.8 Limits Superior and Inferior
6.9 Exercises on limsup and liminf
6.10 Series
6.11 Exercises on Convergence of Series
7 Continuous Functions
7.1 Cauchy's Account of Continuity
7.2 Continuity of the Sine Function
7.3 Limits of Functions
7.4 Exercises on Limits
7.5 The Intermediate Value Theorem
7.6 The Extreme Value Theorem
7.7 Uniform Continuity
7.8 Exercises on Uniform Continuity
7.9 Contraction Mappings and Fixed Points
7.10 A First Look at Permanence
7.11 Exercises on Permanence of Functions
7.12 Sequences of Functions
7.13 Continuity of a Uniform Limit
7.14 Continuity in the Extended Hypersequence
7.15 Was Cauchy Right?
8 Differentiation
8.1 The Derivative
8.2 Increments and Differentials
8.3 Rules for Derivatives
8.4 Chain Rule
8.5 Critical Point Theorem
8.6 Inverse Function Theorem
8.7 Partial Derivatives
8.8 Exercises on Partial Derivatives
8.9 Taylor Series
8.10 Incremental Approximation by Taylor's Formula
8.11 Extending the Incremental Equation
8.12 Exercises on Increments and Derivatives
The Riemann Integral
9.1 Riemann Sums
9.2 The Integral as the Shadow of Riemann Sums
9.3 Standard Properties of the Integral
9.4 Differentiating the Area Function
9.5 Exercise on Average Function Values
10 Topology of the Reals
10.1 Interior, Closure, and Limit Points
10.2 Open and Closed Sets
10.3 Compactness
10.4 Compactness and (Uniform) Continuity
10.5 Topologies on the Hyperreals
III Internal and External Entities
11 Internal and External Sets
11.1 Internal Sets
11.2 Algebra o[ Internal Sets
11.3 Internal Least Number Principle and Induction
11.4 The Overflow Principle
11.5 Internal Order-Completeness
11.6 External Sets
11.7 Defining Internal Sets
11.8 The Underflow Principle
11.9 Internal Sets and Permanence
11.10 Saturation of Internal Sets
11.11 Saturation Creates Nonstandard Entities
11.12 The Size of an Internal Set
11.13 Closure of the Shadow of an Internal Set
11.14 Interval Topology and Hyper-Open Sets
12 Internal Functions and Hyperflnite Sets
12.1 Internal Functions
12.2 Exercises on Properties of Internal Functions
12.3 Hyperfinite Sets
12.4 Exercises on Hyperfiniteness
12.5 Counting a Hyperfinite Set
i2.6 Hyperfinite Pigeonhole Principle
12.7 Integrals as Hyperfinite Sums
IV Nonstandard Frameworks
13 Universes and Frameworks
13.1 What Do We Need in the Mathematical World?
13.2 Pairs Are Enough
13.3 Actually, Sets Are Enough
13.4 Strong Transitivity
13.5 Universes
13.6 Superstructures
13.7 The Language of a Universe
13.8 Nonstandard Frameworks
13.9 Standard Entities
13.10 Internal Entities
13.11 Closure Properties of Internal Sets
13.12 Transformed Power Sets
13.13 Exercises on Internal Sets and Functions
13.14 External Images Are External
13.15 Internal Set Definition Principle
13.16 Internal Function Definition Principle
13.17 Hyperfiniteness
13.18 Exercises on Hyperfinite Sets and Sizes
13.19 Hyperfinite Summation
13.20 Exercises on Hyperfinite Sums
14 The Existence of Nonstandard Entities
14.1 Enlargements
14.2 Concurrence and Hyperfinite Approximation
14.3 Enlargements as Ultrapowers
14.4 Exercises on the Ultrapower Construction
15 Permanence, Comprehensiveness, Saturation
15.1 Permanence Principles
15.2 Robinson's Sequential Lemma
15.3 Uniformly Converging Sequences of Functions
15.4 Comprehensiveness
15.5 Saturation
V Applications
16 Loeb Measure
16.1 Rings and Algebras
16.2 Measures
16.3 Outer Measures
16.4 Lebesgue Measure
16.5 Loeb Measures
16.6 p-Approximability
16.7 Loeb Measure as Approximability
16.8 Lebesgue Measure via Loeb Measure
17 Ramsey Theory
17.1 Colourings and Monochromatic Sets
17.2 A Nonstandard Approach
17.3 Proving Pmsey's Theorem
17.4 The Finite Ramsey Theorem
17.5 The Paris-Harrington Version
17.6 Reference
18 Completion by Enlargement
18.1 Completing the Rationals
18.2 Metric Space Completion
18.3 Nonstandard Hulls
18.4 p-adic Integers
18.5 p-adic Numbers
18.6 Power Series
18.7 Hyperfinite Expansions in Base p
18.8 Exercises
19 Hyperflnite Approximation
19.1 Colourings and Graphs
19.2 Boolean Algebras
19.3 Atomic Algebras
19.4 Hyperfinite Approximating Algebras
19.5 Exercises on Generation of Algebras
19.6 Connecting with the Stone Representation
19.7 Exercises on Filters and Lattices
19.8 Hyperfinite-Dimensional Vector Spaces
19.9 Exercises on (Hyper) Real Subspaces
19.10 The Hahn-Banach Theorem
19.11 Exercises on (Hyper) Linear Functionals
20 Books on Nonstandard Analysis
Index