I will soon be taking a $4$-th year joint undergraduate/graduate level course on Lebesgue Integration and Fourier Analysis, which leads into the next course on Measure and Integration. I generally like to get somewhat comfortable with the material before taking pure math courses, but unlike my previous real/complex analysis classes it hasn't been clear to me from research online what a "standard" reference on these two subjects are.
What is a good text (or texts) which introduce Lebesgue Integration and Fourier Analysis for a strong final year undergraduate?
I know that the subsequent class I shall be taking after Lebesgue Integration and Fourier Analysis (on Measure and Integration) usually uses Folland and Royden as reference texts if that provides any context.
Topics I'd like it to cover (a general idea)
Lebesgue measure on the line, the Lebesgue integral, the monotone and dominated convergence theorems, Lp-spaces, Fourier analysis on the circle, Dirichlet kernel, Riemann-Lebesgue lemma, Fejer's theorem and the convergence of Fourier series, ...
As for my background, I have taken both real and complex analysis, and in our Real Analysis class we covered metric spaces, topological spaces, completeness, the Baire Category Theorem, Urysohn's Lemma, Stone-Weierstraß Theorem, Arzela-Ascoli Theorem... I may be missing a few others. Our reference texts were Rudin (the earlier chapters) and Munkres' book on Topology. I tend to prefer books like Munkres that are slightly more conversational than something like Rudin.
If I can provide any other information please let me know!
From what I've gathered so far, without any further information I think I would give An Introduction to Lebesgue Integration and Fourier Series by Howard Wilcox and David Myers (amazon link) a look.