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

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## A.N. Kolmogorov and S.V. Fomin:

## 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).

### Subject Matter

### 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

- 1. Sets and Functions

- 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

- 5. Basic Concepts

- 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

- 9. Basic Concepts
- 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

- 13. Basic Concepts

- 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

- 18. Continuous Linear Functionals

- 6. LINEAR OPERATORS
- 22. Basic Concepts
- 22.1 Definitions and examples
- 22.2 Continuity and boundedness
- 22.3 Sums and products of operators

- 23. Inverse and Adjoint Operators
- 23.1 The inverse operator. Invertibility
- 23.2 The adjoint operator
- 23.3 The adjoint operator in Hilbert space. Self-adjoint operators
- 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

- 22. Basic Concepts

- 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
- 26.3 Countably additive measures

- 27. Extensions of Measures

- 25. Measure in the Plane

- 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

- 28. Measurable Functions

- 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

- 31. Differentiation of the Indefinite Lebesgue Integral

- 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 $L_2$

- 35. Product Measures. Fubini's Theorem

- Bibliography
- Index

## Source work progress

- 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 3.5$: Well-ordered sets. Ordinal Numbers: Example $2$

- Redoing from start:

- 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $1$ Set Theory: $1$. Sets and Functions: $1.2$: Operations on sets: $(4)$

- Discussion on difference between disjunction and exclusive-or to be analysed