哥德布拉特编著的《超实讲义》是一部讲述非标准分析的入门教程，是由作者数年教学讲义发展并扩充而成。具备基本分析知识的高年级本科生，研究生以及自学人员都可以完全读懂。非标准分析理论不仅是研究无限大和无限小的强有力的理论，也是一种截然不同于标准数学概念和结构的方法，更是新的结构，目标和证明的源泉，推理原理的新起点。书中是从超实数系统开始，从非标准的角度讲述单变量积分，分析和拓扑，着重强调变换原理作为一个重要的数学工具的重要作用。数学宇宙的讲述为全面研究非标准方法论提供了基础保证。最后一章着眼于应用，将这些理论应用于loeb测度理论及其与lebesgue 的一些关系，ramsey 定理，p-进数的非标准结构和幂级数，boolean 代数的stone 表示定理的非标准证明和hahn-banach 定理。《超实讲义》的最大特点尽早引入内集，外集，超有限集，以及集理论扩展方法，较常规的建立在超结构基础上，这样的方式更加显而易见。

读者对象：数学专业的高年级本科生，研究生和科研人员。

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 1.4 Historical References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