## 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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a-algebra absolutely continuous additive set function algebra analogue assume Borel measure Borel set bounded variation Chapter choose closed set completes the proof consider constant continuous on a,b convergence theorem converges uniformly corollary cubes decomposition denote Egorov's theorem equal example Exercise exists fact finite a.e. finite measure formula Fourier coefficients Fourier series Fubini's theorem function defined given Hence If/is implies indefinite integral interval a,b Lebesgue integral Lebesgue measure lemma Let f limsup linear ll/ll measurable functions measurable sets measurable subset measure zero monotone increasing Moreover nonnegative and measurable obtain open intervals open set orthogonal system outer measure partial sums partition prove real-valued respect result follows Riemann integrable Riemann-Stieltjes integral satisfies subinterval Suppose surable tends to zero theorem follows Theorem If f Theorem Let variation on a,b write

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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.