I have taken course in measure theory based from Royden's "Real Analysis" last year and now am reviewing the subject using Folland. However, neither book seems to have many "computational" problems.

Feeling relatively comfortable with the statements of main analysis theorems and results, I would really like to now try working through some more "computational" problems to help me build my intuition about when to use different theorems or when different definitions apply (e.g. solving hard integrals or demonstrating convergence of sequences of integrals using Dominated convergence or Vitali convergence theorems, showing families of functions are uniformly integrable, using Holder's / generalized Holder's / Minkowski / Young's inequalities to solve real integrals, computing Raydon-Nikodym derivatives for real examples, etc).

Question: Are there any good texts / sources of real analysis problems in the flavor of doing actual computations or solving example problems?

I hope this question is not out of place here. I have been studying real analysis for awhile now but still feel that I lack intuition.

Thanks for your help!

  • $\begingroup$ Interesting question. I suspect these abstract theorems are not often useful in computations. I hope you get answers showing how they help prove other theorems in analysis you might consider somewhat less abstract. $\endgroup$ – Ethan Bolker Jun 3 '18 at 17:47
  • $\begingroup$ Last time I opened Schilling’s “Measures, Integrals and Martingales” was more than 2 years ago, so I can’t remember well what kind of problems are there. But this book has a lot of problems (and a solution manual!) $\endgroup$ – User Jun 3 '18 at 19:03

An interesting book which might help to develop the reader's intuition is

  • A Radical Approach to Lebesgue's Theory of Integration

    It provides a historically driven development containing many exercises. With respect to intuition it starts with the following five big questions:

    1. When does a function have a Fouries series expansion that converges to that function?

    2. What is integration?

    3. What is the relationship between integration and differentiation?

    4. What is the relationship between continuity and differentiability?

    5. When can an infinite series be integrated by integrating each term?


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