## Lebesgue Measure and Integration: An IntroductionA superb text on the fundamentals of Lebesgue measure and integration.This book is designed to give the reader a solid understanding of Lebesgue measure and integration. It focuses on only the most fundamental concepts, namely Lebesgue measure for R and Lebesgue integration for extended real-valued functions on R. Starting with a thorough presentation of the preliminary concepts of undergraduate analysis, this book covers all the important topics, including measure theory, measurable functions, and integration. It offers an abundance of support materials, including helpful illustrations, examples, and problems. To further enhance the learning experience, the author provides a historical context that traces the struggle to define "area" and "area under a curve" that led eventually to Lebesgue measure and integration.Lebesgue Measure and Integration is the ideal text for an advanced undergraduate analysis course or for a first-year graduate course in mathematics, statistics, probability, and other applied areas. It will also serve well as a supplement to courses in advanced measure theory and integration and as an invaluable reference long after course work has been completed. |

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This book is a good companion for mathematics students especially the undergraduates.

### Contents

HISTORICAL HIGHLIGHTS | 1 |

11 REARRANGEMENTS | 2 |

12 EUDOXUS 408355 BCE AND THE METHOD OF EXHAUSTION | 3 |

13 THE LUNE OF HIPPOCRATES 430 BCE | 5 |

14 ARCHIMEDES 287212 BCE | 7 |

15 PIERRE FERMAT 16011665 | 10 |

16 GOTTFRIED LEIBNITZ 16461716 ISSAC NEWTON 16421723 | 12 |

17 AUGUSTINLOUIS CAUCHY 17891857 | 15 |

28 CONTINUOUS FUNCTIONS | 66 |

29 DIFFERENTIABLE FUNCTIONS | 73 |

210 SEQUENCES OF FUNCTIONS | 75 |

LEBESGUE MEASURE | 87 |

31 LENGTH OF INTERVALS | 90 |

32 LEBESGUE OUTER MEASURE | 93 |

33 LEBESGUE MEASURABLE SETS | 100 |

34 BOREL SETS | 112 |

18 BERNHARD RIEMANN 18261866 | 17 |

19 EMILE BOREL 18711956 CAMILLE JORDAN 18381922 GIUSEPPE PEANO 18581932 | 20 |

110 HENRI LEBESGUE 18751941 WILLIAM YOUNG 18631942 | 22 |

111 HISTORICAL SUMMARY | 25 |

112 WHY LEBESGUE | 26 |

PRELIMINARIES | 32 |

22 SEQUENCES OF SETS | 34 |

23 FUNCTIONS | 35 |

24 REAL NUMBERS | 42 |

25 EXTENDED REAL NUMBERS | 49 |

26 SEQUENCES OF REAL NUMBERS | 51 |

27 TOPOLOGICAL CONCEPTS OF R | 62 |

35 MEASURING | 115 |

36 STRUCTURE OF LEBESGUE MEASURABLE SETS | 120 |

LEBESGUE MEASURABLE FUNCTIONS | 126 |

42 SEQUENCES OF MEASURABLE FUNCTIONS | 135 |

43 APPROXIMATING MEASURABLE FUNCTIONS | 137 |

44 ALMOST UNIFORM CONVERGENCE | 141 |

LEBESGUE INTEGRATION | 147 |

52 THE LEBESGUE INTEGRAL FOR BOUNDED FUNCTIONS ON SETS OF FINITE MEASURE | 173 |

53 THE LEBESGUE INTEGRAL FOR NONNEGATIVE MEASURABLE FUNCTIONS | 194 |

54 THE LEBESGUE INTEGRAL AND LEBESGUE INTEGRABILITY | 224 |

55 CONVERGENCE THEOREMS | 237 |

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### Common terms and phrases

Borel sets bounded function Calculate Cantor set Cauchy closed set collection construct contains continuous functions converge uniformly countable additivity cr-algebra Definition diIference domain Example fE f fEf+ finite measure finite number follows function defined greatest lower bound Hint Ii(E imfk inverse images least upper bound Lebesgue integral Lebesgue measurable functions Lebesgue measurable sets Lebesgue outer measure lim inf lim sup limit point LMCT mathematics measure zero monotone increasing mutually disjoint natural number nondecreasing nonempty set nonmeasurable nonnegative measurable functions nonnegative simple functions numbers in 0,1 open cover open intervals open set partition Problem Proof Proposition 5.2 Proposrr1on rational numbers reader may show real numbers real-valued function Riemann integrable sequence of measurable sequence of sets sequence of simple set of measure set of real show lim sigma algebra step function subintervals ternary expansion uncountable uniform convergence