The theory of functions of a real variable /

This textbook leads the reader by easy stages through the essential parts of the theory of sets and theory of measure to the properties of the Lebesgue integral.

Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Jeffery, R. L.
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Toronto : University of Toronto Press, 1951.
Colección:Mathematical expositions ; no. 6.
Temas:
Functions of real variables.
Fonctions de variables réelles.
MATHEMATICS > Calculus.
MATHEMATICS > Mathematical Analysis.
Functions of real variables
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Cover
  • CONTENTS
  • INTRODUCTION
  • 0.1. The positive integers
  • 0.2. The fundamental operations on integers
  • 0.3. The rational numbers
  • 0.4. The irrational numbers
  • 0.5. The real number system
  • Problems
  • I. SETS, SEQUENCES, AND FUNCTIONS
  • 1.1. Bounds and limits of sets and sequences
  • 1.2. Functions and their properties
  • 1.3. Sequences of functions and uniform convergence
  • Problems
  • II. METRIC PROPERTIES OF SETS
  • 2.1. Notation and definitions
  • 2.2. Descriptive properties of sets
  • 2.3. Metric properties of sets
  • 2.4. Measurability and measurable sets2.5. Further descriptive properties of sets
  • 2.6. Measure-preserving transformations and non-measurable sets
  • 2.7. A non-measurable set
  • Problems
  • III. THE LEBESGUE INTEGRAL
  • 3.1. Measurable functions
  • 3.2. The Lebesgue integral
  • 3.3. The Riemann integral
  • 3.4. The extension of the definition of the Lebesgue integral to unbounded functions
  • 3.5. Further properties of measurable functions
  • Problems
  • IV. PROPERTIES OF THE LEBESGUE INTEGRAL
  • 4.1. Notation and conventions
  • 4.2. Properties of the Lebesgue integral4.3. Definitions of summability and their extension to unbounded sets
  • 4.4. The integrability of sequences
  • 4.5. Integrals containing a parameter
  • 4.6. Further theorems on sequences of functions
  • 4.7. The ergodic theorem
  • Problems
  • V. METRIC DENSITY AND FUNCTIONS OF BOUNDED VARIATION
  • 5.1. The Vitali covering theorem
  • 5.2. Metric density of sets
  • 5.3. Approximate continuity
  • 5.4. Functions of bounded variation
  • 5.5. Upper and lower derivatives
  • 5.6. Functions of sets
  • 5.7. The summability of the derivative of a function of bounded variation5.8. Functions of sets
  • Problems
  • VI. THE INVERSION OF DERIVATIVES
  • 6.1. Functions defined by integrals, F(x) = L(f, a, x)
  • 6.2. The inversion of derivatives which are not summable
  • 6.3. The integrals of Denjoy and other generalized integrals
  • 6.4. Descriptive definitions of generalized integrals
  • Problems
  • VII. DERIVED NUMBERS AND DERIVATIVES
  • 7.1. Derivatives or derived numbers
  • 7.2. The Weierstrass non-differentiable function
  • 7.3. A function which has no unilateral derivative7.4. The derived numbers of arbitrary functions defined on arbitrary sets
  • 7.5. Approximate derived numbers over arbitrary sets
  • 7.6. Approximate derived numbers of measurable functions, and relations between arbitrary functions and measurable functions
  • VIII. THE STIELTJES INTEGRAL
  • 8.1. The Riemann-Stieltjes Integral
  • 8.2. Properties of the Riemann-Stieltjes integral
  • 8.3. Interval functions and measure functions
  • 8.4. Linear functionals
  • BIBLIOGRAPHY
  • INDEX OF SUBJECTS
  • A
  • B
  • C
  • D

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