Real AnalysisThis is the classic introductory graduate text. Heart of the book is measure theory and Lebesque integration. 
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Review: Real Analysis
User Review  Thomas  GoodreadsIf I can, I would give it one and a half. I read only seven chapters of the book.The merits are that it is a slow introduction to Lebesgue measure and integration. On the other hand, a lot of non ... Read full review
Review: Real Analysis
User Review  GoodreadsIf I can, I would give it one and a half. I read only seven chapters of the book.The merits are that it is a slow introduction to Lebesgue measure and integration. On the other hand, a lot of non ... Read full review
Contents
Prologue to the Student  1 
The Real Number System  31 
The Lebesgue Integral  75 
Copyright  
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Common terms and phrases
aalgebra afinite absolutely continuous axiom Baire sets Banach space Borel measure Borel sets called Cauchy sequence closed set cluster point compact Hausdorff space compact set complete metric space continuous function continuous map continuous realvalued function Convergence Theorem Corollary countable collection cralgebra crfinite definition denote dense element equicontinuous finite measure finite number following proposition func function defined function f given Hausdorff space Hence Hint homeomorphism inequality inner regular integrable function isomorphic Lebesgue measure Lemma Let f Let G locally compact Hausdorff measurable function measurable sets measure space measure zero monotone natural numbers nonempty nonnegative measurable function onetoone open covering open intervals open set open subset outer measure point of closure Problem Proof Prove Proposition satisfies sequence of measurable set function set of measure Show simple functions subspace tion topological space topologically equicontinuous unique