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Prologue to the Student | 1 |
Topological Spaces | 4 |
Measure and Outer Measure | 12 |
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absolutely continuous axiom Baire measure Baire set Banach space Borel measure Borel sets bounded linear functional called Cauchy sequence closed sets compact Hausdorff space compact space continuous function continuous real-valued function Convergence Theorem countable collection definition denote E₁ element finite measure finite number fn(x following proposition function defined function f ƒ and g given Hausdorff space Hence homeomorphism integrable function L₁ Lebesgue measure Lemma Let f Let ƒ locally compact measurable function measurable sets measure space measure zero monotone natural numbers nonempty nonnegative measurable function O₁ one-to-one open intervals open set outer measure point of closure Problem Proof rational numbers semicontinuous sequence fn sequence of measurable set containing set function set of finite set of measure Show simple functions subspace topological space topological vector space unique