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Personal favorites, in suggested reading order:-Bartle,

  • Bartle, The Elements of Integration. Exercises are all doable and at about the same level. The best first-time text.-Royden.

  • Royden. I loved this book: the exercises vary from easy to quite tricky (when you need to sleep on it!or take a looong shower) and working as many exercises as possible, especially the hard ones, is a great way to really understand the real numbers.-Rudin.

  • Rudin. The proofs are at times way too slick, formulas popping out with no motivation, but that should be no real (or complex?) problem after Royden, and it gives a beautiful overview of the essentials; the topology part is great too. The problems are great and often quite challenging.-Oxtoby,

  • Oxtoby, Measure and Category. This is just a fantastic little book. After you have studied the others, you can read through this like a novel and everything will start to fit together much more. Pure inspiration.-

  • Dunford and Schwarz. Some encounter with this is necessary, especially after you've also been through Rudin's Functional Analysis.-

  • Lamperti's "Probability". This could be called "Probability for Analysts" and it is a beautiful little book. -

  • Billingsley's Ergodic Theory and Information. Now you're ready to see what some of that abstract stuff is good for, and this beautiful text is an excellent choice.
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Personal favorites, in suggested reading order: -Bartle, The Elements of Integration. Exercises are all doable and at about the same level. The best first-time text. -Royden. I loved this book: the exercises vary from easy to quite tricky (when you need to sleep on it!or take a looong shower) and working as many exercises as possible, especially the hard ones, is a great way to really understand the real numbers. -Rudin. The proofs are at times way too slick, formulas popping out with no motivation, but that should be no real (or complex?) problem after Royden, and it gives a beautiful overview of the essentials; the topology part is great too. The problems are great and often quite challenging. -Oxtoby, Measure and Category. This is just a fantastic little book. After you have studied the others, you can read through this like a novel and everything will start to fit together much more. Pure inspiration. -Dunford and Schwarz. Some encounter with this is necessary, especially after you've also been through Rudin's Functional Analysis. -Lamperti's "Probability". This could be called "Probability for Analysts" and it is a beautiful little book. -Billingsley's Ergodic Theory and Information. Now you're ready to see what some of that abstract stuff is good for, and this beautiful text is an excellent choice.