# Book:A.N. Kolmogorov/Introductory Real Analysis

## A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis

Published $\text {1968}$, Dover Publications

ISBN 0-486-61226-0 (translated by Richard A. Silverman)

Analysis
Metric Spaces
Topology
Linear Algebra
Measure Theory
Calculus

### Contents

Editor's Preface (Richard A. Silverman)
1. SET THEORY
1. Sets and Functions
1.1 Basic definitions
1.2 Operations on sets
1.3 Functions and mappings. Images and preimages
1.4 Decomposition of a set into classes. Equivalence relations
2. Equivalence of Sets. The Power of a Set
2.1 Finite and infinite sets
2.2 Countable sets
2.3 Equivalence of sets
2.4 Uncountability of the real numbers
2.5 The power of a set
2.6 The Cantor-Bernstein theorem
3. Ordered Sets and Ordinal Numbers
3.1 Partially ordered sets
3.2 Order-preserving mappings. Isomorphisms
3.3 Ordered sets. Order types
3.4 Ordered sums and products of ordered sets
3.5 Well-ordered sets. Ordinal numbers
3.6 Comparison of ordinal numbers
3.7 The well-ordering theorem, the axiom of choice and equivalent assertions
3.8 Transfinite induction
3.9 Historical remarks
4. Systems of Sets
4.1 Rings of sets
4.2 Semirings of sets
4.3 The ring generated by a semiring
4.4 Borel algebras

2. METRIC SPACES
5. Basic Concepts
5.1 Definitions and examples
5.2 Continuous mappings and homeomorphisms. Isometric spaces
6. Convergence. Open and closed sets
6.1 Closure of a set. Limit points
6.2 Convergence and limits
6.3 Dense subsets. Separable spaces
6.4 Closed sets
6.5 Open sets
6.6 Open and closed sets on the real line
7. Complete metric spaces
7.1 Definitions and examples
7.2 The nested sphere theorem
7.3 Baire's theorem
7.4 Completion of a metric space
8. Contraction mappings
8.1 Definition of a contraction mapping. The fixed point theorem
8.2 Contraction mappings and differential equations
8.3 Contraction mappings and integral equations

3. TOPOLOGICAL SPACES
9. Basic Concepts
9.1 Definitions and examples
9.2 Comparison of topologies
9.3 Bases. Axioms of countability
9.4 Convergent sequences in a topological space
9.5 Axioms of separation
9.6 Continuous mappings. Homeomorphisms
9.7 Various ways of specifying topologies. Metrizability
10. Compactness
10.1 Compact topological spaces
10.2 Continuous mappings of compact spaces
10.3 Countable compactness
10.4 Relatively compact subsets
11. Compactness in Metric Spaces
11.1 Total boundedness
11.2 Compactness and total boundedness
11.3 Relatively compact subsets of a metric space
11.4 Arzelà's Theorem
11.5 Peano's Theorem
12. Real Functions on Metric and Topological Spaces
12.1 Continuous and uniformly continuous functions and functionals
12.2 Continuous and semicontinuous functions on compact spaces
12.3 Continuous curves in metric spaces
4. LINEAR SPACES
13. Basic Concepts
13.1 Definitions and examples
13.2 Linear dependence
13.3 Subspaces
13.4 Factor spaces
13.5 Linear functionals
13.6 The null space of a functional. Hyperplanes
14. Convex Sets and Functionals. The Hahn-Banach Theorem
14.1 Convex sets and bodies
14.2 Convex functionals
14.3 The Minkowski functional
14.4 The Hahn-Banach theorem
14.5 Separation of convex sets in a linear space
15. Normed Linear Spaces
15.1 Definitions and examples
15.2 Subspaces of a normed linear space
16. Euclidean Spaces
16.1 Scalar products. Orthogonality and bases
16.2 Examples
16.3 Existence of an orthogonal basis. Orthogonalization
16.4 Bessel's inequality. Closed orthogonal systems
16.5 Complete Euclidean spaces. The Riesz-Fischer theorem
16.6 Hilbert space. The isomorphism theorem
16.7 Subspaces. Orthogonal complements and direct sums
16.8 Characterization of Euclidean spaces
16.9 Complex Euclidean spaces
17. Topological Linear Spaces
17.1 Definitions and examples
17.2 Historical remarks

5. LINEAR FUNCTIONALS
18. Continuous Linear Functionals
18.1 Continuous linear functionals on a topological linear space
18.2 Continuous linear functionals on a normed linear space
18.3 The Hahn-Banach theorem for a normed linear space
19. The Conjugate Space
19.1 Definition of the conjugate space
19.2 The conjugate space of a normed linear space
19.3 The strong topology in the conjugate space
19.4 The second conjugate space
20. The Weak Topology and Weak Convergence
20.1 The weak topology in a topological linear space
20.2 Weak convergence
20.3 The weak topology and weak convergence in a conjugate space
20.4 The weak* topology
21. Generalized Functions
21.1 Preliminary remarks
21.2 The test space and test functions. Generalized functions
21.3 Operations on generalized functions
21.4 Differential equations and generalized functions
21.5 Further developments

6. LINEAR OPERATORS
22. Basic Concepts
22.1 Definitions and examples
22.2 Continuity and boundedness
22.3 Sums and products of operators
23.1 The inverse operator. Invertibility
23.4 The spectrum of an operator. The resolvent
24. Completely Continuous Operators
24.1 Definitions and examples
24.2 Basic properties of completely continuous operators
24.3 Completely continous operators in Hilbert space

7. MEASURE
25. Measure in the Plane
25.1 Measure of elementary sets
25.2 Lebesgue measure of plane sets
26. General Measure Theory
26.1 Measure on a semiring
27. Extensions of Measures

8. INTEGRATION
28. Measurable Functions
28.1 Basic properties of measurable functions
28.2 Simple functions. Algebraic operations on measurable functions
28.3 Equivalent functions
28.4 Convergence almost everywhere
28.5 Egorov's theorem
29. The Lebesgue Integral
29.1 Definition and basic properties of the Lebesgue integral
29.2 Some key theorems
30. Further Properties of the Lebesgue Integral
30.1 Passage to the limit in Lebesgue integrals
30.2 The Lebesgue integral over a set of infinite measure
30.3 The Lebesgue integral vs. the Riemann integral

9. DIFFERENTIATION
31. Differentiation of the Indefinite Lebesgue Integral
31.1 Basic properties of monotonic functions
31.2 Differentiation of a monotonic function
31.3 Differentiation of an integral with respect to its upper limit
32. Functions of a Bounded Variable
33. Reconstruction of a Function from its Derivative
33.1 Statement of the problem
33.2 Absolutely continuous function
33.3 The Lebesgue decomposition
34. The Lebesgue Integral as a Set Function
34.1 Charges. The Hahn and Jordan decompositions
34.2 Classification of charges. The Radon-Nikodým theorem

10. MORE ON INTEGRATION
35. Product Measures. Fubini's Theorem
35.1 Direct products of sets and measures
35.2 Evaluation of a product measure
35.3 Fubini's theorem
36. The Stieltjes Integral
36.1 Stieltjes measure
36.2 The Lebesgue-Stieltjes integral
36.3 Applications to probability theory
36.4 The Riemann-Stieltjes integral
36.5 Helly's theorems
36.6 The Riesz representation theorem
37. The spaces $L_1$ and $L_2$
37.1 Definition and basic properties of $L_1$
37.2 Definition and basic properties of $L_2$

Bibliography
Index

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## Source work progress

Redoing from start:
Discussion on difference between disjunction and exclusive-or to be analysed