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I have taken a first course in real analysis and I'm currently studying analysis in $\mathbb{R}^N$ on my own. I want to start functional analysis after this, and I also want to learn measure theory and Lebesgue integration.

My question is: should I learn functional analysis first, without Lebesgue integration, using a text such as Kreyszig's introductory functional analysis, and then study Lebesgue integration later, or should I study Lebesgue theory first and then functional analysis, perhaps using using something like Lang's "Real and Functional Analysis"? If neither of these ways is good in your opinion, what would be the best way to go?

I'm a grad student of mechanical engineering, and I ultimately want to understand PDEs properly.

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It's certainly good to know about Lebesgue integration, but if you're willing to accept a few properties of measures, you can get by without a deep knowledge of Lebesgue integration right away. And that approach appears to be compatible with Kreyszig's book.

If you were going to switch into Mathematics, then I might suggest using Lang's book and going that route. But with your background, I think Kreyszig is a much better choice. For one thing, you'll learn useful tools for Engineering from Kreyszig, and those tools will probably benefit you immediately and in several ways. For another, Kreyszig has written a classic book on Applied Math and knows how to direct your study in a goal-oriented way that is general, illuminating, and productive.

Presumably your desire to better understand Engineering is what has brought you to this point, and I would not let anything sidetrack you too much along the way. Mathematics can be seductive in the purity of its approach, but keeping on a directed path without too much wandering seems wise and--in the long run--far more satisfying. Keep your eye on the ball.

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    $\begingroup$ Solid advice, T.A.E. $\endgroup$ Commented Dec 10, 2014 at 23:48
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    $\begingroup$ I second this advice, even for a pure mathematician. I learned my earliest functional analysis a full year before I even touched measure theory. For $L^p$-spaces, pretend they're Riemann integrable functions with an intuitive notion for the "almost everywhere" caveat and you'll do fine there. Seeing some functional analysis before rigorous measure theory helped me view Lebesgue integration from multiple contexts, which I found extremely valuable. $\endgroup$ Commented Dec 10, 2014 at 23:57
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    $\begingroup$ "Mathematics can be seductive in the purity of its approach.." I've already experienced that! Well, thanks, @T.A.E., neuguy, Lebron. I guess I'll return to measure theory someday. $\endgroup$ Commented Dec 12, 2014 at 3:16
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    $\begingroup$ @T.A.E. One more thing. I came across Karen Saxe's Beginning Functional Analysis. It contains a one-chapter intro to Lebesgue integration and then uses it in developing functional analysis. What do you think about such an approach? $\endgroup$ Commented Dec 12, 2014 at 3:18
  • $\begingroup$ @AdityaKashi : I think the Chapter on Lebesgue Integration would be enough for $\mathbb{R}^{n}$, and wouldn't take you too far off into left field. $\endgroup$ Commented Dec 12, 2014 at 5:35

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