About Lebesgue integration...
I want to apply Lebesgue integration-based theorems in Fourier analysis for functions that are not integrable by Riemann.
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M$2 Answers
To do what you seek to do (i.e., apply Fourier-theroetic theorems to functions that are not Reimann integrable), it might do you well to identify characteristics of functions that are Lebesque integrable but not Reimann integrable, e.g., but reviewing in the book 'Counterexamples in Analysis' by Gelbaum and Almstead.
Personal experience
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M$Concerning set theory, some basic topology knowledge should be enough. (open/closed sets, compact topological spaces, topological vector spaces...)
http://en.wikipedia.org/wiki/Lebesgue_integration
Armstrong, M.A. Basic Topology (http://www.springer.com/math/geometry/book/978-0-387-90839-7)
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M$
I let the question expire since both were good, I didn't think it would be fair to select a better one myself.
I do lean towards the book reference and a more personal answer, however.