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Chapters 1-3, 6 and 7 of
Folland's Real Analysis are very good. Another good source are
chapters 1-4, 6, 11 and 12 of Royden's Real Analysis.
(Chapters 1-4 on Lebesgue measure should be read by everybody.) A
more sophisticated treatment can be found in chapters 1-3 and 8 of
Rudin's Real & Complex Analysis. See also Chapter 10 of Stoll's book on real