## Lebesgue's Theory of Integration: Its Origins and DevelopmentIn this book, Hawkins elegantly places Lebesgue's early work on integration theory within in proper historical context by relating it to the developments during the nineteenth century that motivated it and gave it significance and also to the contributions made in this field by Lebesgue's contemporaries. Hawkins was awarded the 1997 MAA Chauvenet Prize and the 2001 AMS Albert Leon Whiteman Memorial Prize for notable exposition and exceptional scholarship in the history of mathematics. |

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### Contents

Introduction | ix |

Riemanns Theory of Integration | 3 |

The Development of Riemanns Ideas 187080 | 21 |

Set Theory and the Theory of Integration | 55 |

The End of the Century A Period of Transition | 86 |

The Creation of Modern Integration Theory | 120 |

Pioneering Applications of the Lebesgue Integral | 163 |

The LebesgueStieltjes Integral | 179 |

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### Common terms and phrases

absolutely continuous arbitrary Arzela Bois-Reymond Borel bounded function bounded variation Cantor Cauchy Cauchy's coefficients concept considered continuous functions converges corresponding countable curve length Darboux denote dense set differentiable Dini derivatives Dini's Dirichlet's discontinuous functions equation everywhere example exists extended Fatou finite number Fourier series Fubini's Theorem func functions of bounded Fundamental Theorem greatest lower bound Hankel Harnack ideas infinite integrable functions Jordan least upper bound Lebesgue integral Lebesgue measure Lebesgue's Theorem lemma Math mathematicians measurable sets measure zero measure-theoretic notion number of intervals number of points Osgood paper partition Peano points of discontinuity problem proof of Theorem proved Radon real numbers rectifiable curve result Riemann integral Riemann's definition Riemann's theory Riesz Schoenflies sequence set of points showed species sets Stieltjes subset term-by-term integration theory of integration theory of measure tion trigonometric series unbounded uniform convergence uniformly validity Vitali Volterra Weierstrass