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I'm looking for a book on complex analysis that has a similar writing style to either Terry Tao's Analysis II or Nathan Jacobson's Basic Algebra series. I have found both of these extremely easy to read and follow the proofs given quite easily, or at least I have so far. Preferably the level would be for an advanced undergraduate course or a book used in a first year graduate class on complex analysis.

For the measure theory book, I was wondering what a good book would be for a study of the subject starting from almost the ground level (like why Riemann's criteria for integrability is not entirely sufficient), but would be accessible to an advanced undergraduate.

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    $\begingroup$ -1 This should really be two questions. $\endgroup$ – Michael Greinecker Jun 26 '14 at 10:00
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For complex analysis, I would recommend Marsden and Hoffman's Basic Complex Analysis. Very detailed and shows every step, and it's a good introduction for those not ready for Rudin (Real and Complex anaylsis). As for measure theory, baby Rudin (Principles of Mathematical Analysis) is undoubtedly the standard to which all others should be held. The text starts by proving most of the theorems used in calculus and culminates with great chapters on measure theory and the Lebesgue theory of integration.

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For Measure theory :

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For complex analysis there is a beautiful book by Stein and Shakarchi. I would recommend it very strongly. It starts at an elementary level and takes the reader to advanced graduate level topics. The exposition is lucid and the exercises are good.

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