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I've been reading Larson and Bengzon's recent book on finite element methods, which has been good for getting an understanding of basic theory and computational procedures. The finite element book by Brenner and Scott has been recommended strongly by a number of people as a book for understanding more of the theory behind FEM.

In the introduction, the only prerequisite mentioned is "a course in real variables"; however, the proofs in the book seem to draw heavily from a measure theoretic treatment of the Lebesgue integral not necessarily covered in a standard two-course real analysis sequence. (For instance, exercises in Chapter 1 suggest using the monotone convergence theorem and Fubini's theorem.) Having taken a couple courses on real analysis that used Riemann integration and Jordan measure, plus a course on functional analysis that developed Lebesgue integration without fully developing the Lebesgue measure, I have proven related versions of those theorems (years ago) without using Lebesgue measure or measure theory, but it's been a while.

To do and understand the proofs of the FEM convergence theory in Brenner and Scott, how critical is it to really understand Lebesgue measure theory (say, at the level of Adams and Guillemin)?

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I don't believe you need any measure theory, just enough integration theory to make sense of Lebesgue and Sobolev spaces. If you know the statements of dominated convergence and Fubini's theorem as well as the fundamental lemma of the calculus of variations ($\int u \phi \,dx = 0$ for all $\phi\in C^\infty_0$ implies $u=0$), you should be fine. Chapter 1 is somewhat special since it is a review of these spaces, so the corresponding exercises are more fundamental as well. The remaining book is on a higher level (but does assume you know some functional analysis and the theory of (weak solutions to) partial differential equations).

(For what it's worth, I found Braess' book, which also treats the mathematical theory of finite elements, to be a friendlier read, and it might be a better place to start.)

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