## Measure and Integral: An Introduction to Real AnalysisThis volume develops the classical theory of the Lebesgue integral and some of its applications. The integral is initially presented in the context of n-dimensional Euclidean space, following a thorough study of the concepts of outer measure and measure. A more general treatment of the integral, based on an axiomatic approach, is later given. Closely related topics in real variables, such as functions of bounded variation, the Riemann-Stieltjes integral, Fubini's theorem, L(p)) classes, and various results about differentiation are examined in detail. Several applications of the theory to a specific branch of analysis--harmonic analysis--are also provided. Among these applications are basic facts about convolution operators and Fourier series, including results for the conjugate function and the Hardy-Littlewood maximal function. Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis for student interested in mathematics, statistics, or probability. Requiring only a basic familiarity with advanced calculus, this volume is an excellent textbook for advanced undergraduate or first-year graduate student in these areas. |

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### Contents

Introduction Chapter 1 Preliminaries 1 Points and Sets in | 1 |

R as a Metric Space | 2 |

Open and Closed Sets in R Special Sets | 5 |

Compact Sets the HeineBorel Theorem | 8 |

Functions | 9 |

Continuous Functions and Transformations | 10 |

The Riemann Integral Exercises | 11 |

5 | 12 |

Two Properties of Lebesgue Measure | 40 |

Characterizations of Measurability | 42 |

Lipschitz Transformations of R | 44 |

A Nonmeasurable Set | 46 |

Exercises | 47 |

Lebesgue Measurable Functions | 50 |

The Lebesgue Integral | 64 |

Repeated Integration | 87 |

11 | 13 |

Functions of Bounded Variation the Riemann Stieltjes Integral | 15 |

Rectifiable Curves | 21 |

The RiemannStieltjes Integral | 23 |

Further Results About RiemannStieltjes Integrals | 28 |

Exercises | 31 |

Lebesgue Measure and Outer Measure | 33 |

Lebesgue Measurable Sets | 37 |

Differentiation | 98 |

IS Classes | 125 |

Approximations of the Identity Maximal Functions | 145 |

Abstract Integration | 161 |

Outer Measure Measure | 193 |

A Few Facts From Harmonic Analysis | 211 |

Notation | 265 |

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Page ix - ... properties (i) ||x|| > 0 and ||x|| = 0 if and only if x = 0. (ii) ||ax|| = a| ||x||, where |a| is the absolute value of the complex scalar a.