I'm an undergraduate with somewhat little analysis background (one undergraduate course a year ago and a graduate complex analysis course this semester) who wishes to self-study advanced real analysis (measure theory, Lebesgue integration, etc.). This is most immediately so that I can take a Fourier analysis course next semester (textbook: Fourier Analysis by Duoandikoetxea, if anyone is familar), but I also just because I hope to get a deep and meaningful understanding of the topic for its own purposes. I've been reading Integral, Measure, and Derivative: a Unified Approach by Shilov and was wondering if anyone could comment on its ultimate utility as a first introduction to the topic.
After a few days of reading, I've finished chapter 3 (n-space Lebesgue integration) and feel I understand everything presented in the book well enough, but any time I look at an outside source I become completely lost. The Daniell scheme definitions rarely match with what nearly everyone else uses, and while I'm sure Shilov will eventually prove them equivalent, I can't help but feel like I'm making everything harder for myself by using this book. For example, when I think of a measurable function on a certain set, I think of a function which is the almost-everywhere limit of a sequence of some of the elementary functions which define integration on that set, but when I Google it, I'm hit with some measure theory-based definition that is completely new to me. Is the generality of the Daniell scheme worth being this (temporarily) isolated or should I look to switch to another text and only later come back to Shilov?
I own but have not read Kolmogorov's Elements of the Theory of Functions and Functional Analysis, if anyone has an opinion to share on that. I was also thinking of potentially buying Stein's Princeton Lectures in Analysis edition. Any other suggestions/reviews would be appreciated, too.