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Real Analysis - Royden, Halsey, and Fitzpatrick, Patrick
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Dealing with measure theory and Lebesque integration, this is an introductory graduate text.

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Real Analysis 2010, Prentice Hall, Upper Saddle River, NJ

ISBN-13: 9780131437470

4th edition

Hardcover

Real Analysis 1988, Pearson, New York, NY

ISBN-13: 9780024041517

3rd edition

Hardcover