## 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 6 Continuous Functions and Transformations | 10 |

The Riemann Integral Exercises | 11 |

5 | 12 |

11 | 13 |

Characterizations of Measurability | 42 |

Lipschitz Transformations of R | 44 |

A Nonmeasurable Set | 46 |

Exercises | 47 |

Lebesgue Measurable Functions | 50 |

Egorovs | 56 |

The Lebesgue Integral | 64 |

Repeated Integration | 87 |

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 |

Two Properties of Lebesgue Measure | 40 |

### Common terms and phrases

a-algebra a-finite absolutely continuous additive set function ak,bk algebra analogue assume Borel measure Borel set bounded variation Chapter choose completes the proof consider constant continuous on a,b convergence theorem converges uniformly corollary cubes decomposition denote equal example Exercise exists fact finite measure formula Fourier coefficients Fourier series function defined given Hence Holder's inequality If/is implies indefinite integral interval a,b kernel last integral Lebesgue integral Lebesgue measure Lebesgue point Lebesgue-Stieltjes lemma Let f limsup linear ll/ll measurable functions measurable sets measurable subsets measure space measure zero monotone increasing Moreover nonnegative and measurable obtain open set opposite inequality orthogonal system outer measure partial sums partition properties real-valued Riemann integrable Riemann-Stieltjes integral satisfies sequence singular subinterval summable Suppose surable tends to zero theorem follows Theorem Let trigonometric variation on a,b write

### Popular passages

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.