I'm studying real&complex analysis, functional analysis, measure theory, and Lebesgue integral. I want to make sure which books to use in order to study these fields most rigorously and deeply.
Regarding analysis, I have read the following books and several other less rigorous books which are not listed here: Baby Rudin, Apostol Calculus Vol. 1, Introductory Real Analysis by Kolmogorov, Complex Analysis by Ahlfors, Calculus on Manifords by Spivak
and I'm considering to use the following books: Real Analysis by Carothers, Real Analysis by Royden, Functional Analysis by Rudin, Real and Complex Analysis by Rudin, Adams Measure Theory and Probability.
Is going over Carother's book at this moment waste of time? Or is it worthwhile due to its rich problem sets? Is there any flaw in the list of the books which I'm currently considering to use? Which book do you recommend me to read?