## Real Analysis |

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### Contents

Set Theory I | 1 |

THEORY OF FUNCTIONS OF A REAL VAR1ABLE | 19 |

Lebesgue Measure | 43 |

Copyright | |

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### Common terms and phrases

a-algebra a-finite absolutely continuous axiom Baire measure Banach space Borel equivalent Borel measure Borel sets Borel subset bounded linear functional called Cauchy sequence closed sets cluster point compact Hausdorff space compact space complete continuous function continuous mapping continuous real-valued functions convex set Corollary countable collection countably compact Daniell integral definition denote elements finite measure finite number following proposition function defined function f given Hausdorff space Hence homeomorphism implies infinite sequence Lebesgue measure Lemma Let f Let/be lim xn linear manifold locally compact measurable function measurable sets measure algebra measure space measure zero metric space monotone natural numbers nonempty nonnegative measurable function normed linear space notion one-to-one open intervals open set outer measure point of closure Problem Proof rational numbers semicontinuous set function set of finite set of measure set of real Show simple function subspace suppose topological space unique