## 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

Chapter | 1 |

Chapter | 9 |

Stieltjes Integral | 15 |

i | 33 |

12 | 48 |

Lebesgue Measurable Functions | 50 |

23 | 60 |

The Lebesgue Integral | 64 |

LP Classes | 125 |

Approximations of the Identity Maximal Functions | 145 |

The HardyLittlewood Maximal Function | 155 |

Abstract Integration | 161 |

Measurable Functions Integration | 167 |

Absolutely Continuous and Singular | 174 |

The Dual Space of L | 182 |

Outer Measure Measure | 193 |

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absolutely continuous additive apply assume Borel bounded variation called Chapter choose Clearly closed collection compact completes the proof condition consider constant construction containing converges Corollary countable covering cubes defined definition denote disjoint equal everywhere example Exercise exists f and g f is continuous f is measurable fact finite fixed follows formula Fourier function f given gives Hence holds implies increasing inequality interval Lebesgue integral lemma Let f limit limsup linear means measurable functions measurable sets measurable subset measure zero monotone Moreover Note obtain outer measure particular partition positive prove relation relative respect result Riemann integrable satisfies sequence simple space subset Suppose taking tends Theorem Let union 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.