I've just graduated in physics and I intend to do a master's and doctorate in mathematical physics, but I don't have anyone to guide me in this path and I'm studying as a self-taught. When looking into the math needed for this with an expert in the field he told me that I would need to know functional analysis, topology and probability (it would be something on the levels of Reed and Simons, Munkres and Varadhan respectively). It's just that I left a course with a weak mathematical foundation and I'm reviewing linear algebra and calculus, and when researching more about functional analysis, topology and probability I realized that to understand well I need to have deep knowledge of analysis, real analysis, metric spaces, measure theory and maybe even other content that I don't even know exists yet. So I'm here looking for a recommendation for an analysis book that I can study on my own that covers everything from zero on analysis to advanced real analysis and explains metric spaces and measure theory. Is there any book or set of books that can help me with this?
I already bought some analysis books to study based on colleagues' comments but I still don't feel safe, some of them are:
-Principles of Mathematical Analysis and Real and Complex Analysis: Affordable Edition by Walter Rudin;
-Real Analysis 1, 2 and 3 and Analysis Course 1 and 2 by Elon Lages Lima;
-Analysis I by Djairo Guedes de Figueredo;
-Real Analysis - Functions of One Real Variable of Bourchtein and Bourchtein;
-Mauritius Zahn's Real Analysis;
-Introduction to Mathematical Analysis by Geraldo Ávila.
The way a math book is written for math students might not make it suitable for physics students. In any case, take a look at Appel's Mathematics for Physics and Physicists.