Intermediary self-learning-readable book(s) for Real Analysis (incl. Measure Theory,…)?

Question

After studying a very readable book, Advanced Calculus by Fitzpatrick, I thought I start more advanced of real analysis by the same author so I started Real Analysis by Fitzpatrick (and Royden). Well, first chapter is easy since I have studied same things from Fitzpatrick's Advanced Calculus; however, from the beginning of the second chapter the topics are completely new and it introduces and explains as if the reader know about it beforehand (i.e. with many gaps and not readable as it was for Fitzpatrick's Advanced Calculus). By the way, the second chapter's topics are:

2.1 Introduction

2.2 Lebesgue Outer Measure

2.3 The a-Algebra of Lebesgue Measurable Sets

2.4 Outer and Inner Approximation of Lebesgue Measurable Sets

2.5 Countable Additivity, Continuity, and the Borel-Cantelli Lemma

2.6 Nonmeasurable Sets

2.7 The Cantor Set and the Cantor-Lebesgue Function

Could somebody let me know book(s) easier to read for self learning for these topics, or for real analysis in general which includes same topics of whole book of Fitzpatrick's Real Analysis which are in here?

Answer 1

I really like Terence Tao's An introduction to Measure Theory (which is available from his website) but it has no contents related to Hilbert spaces, weak and *-weak convergence or anything related to functional analysis. You can also consider Folland's Real Analysis for this part.

For the functional analysis part I would recommend you Lax's Functional Analysis.

You can also take a look at this: https://mathoverflow.net/questions/72419/a-good-book-of-functional-analysis