## Real analysis |

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### Contents

Prologue to the Student | 1 |

The Real Number System | 29 |

The Lebesgue Integral | 73 |

Copyright | |

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absolutely continuous Baire measure Baire set Banach space Borel equivalence Borel measure Borel sets Borel subset bounded linear functional called Cauchy sequence closed sets cluster point compact Hausdorff space compact space continuous function continuous real-valued function Convergence Theorem countable collection countably compact definition denote element finite measure finite number following proposition function defined function f given Hausdorff space Hence homeomorphism If/is implies integrable function Lebesgue measure Lemma Let f Let/be lim xn locally compact measurable function measurable sets measure space measure zero monotone natural numbers nonempty nonnegative measurable function normed linear space one-to-one open intervals open set outer measure point of closure Problem Proof r-algebra containing r-finite rational numbers semicontinuous sequence of measurable sequence xn set containing set function set of finite set of measure set of real Show simple functions subspace topological space topological vector space unique