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This is an attempt to get a list going of books in mathematics (algebra, topology, geometry etc.) that in the spirit of Terence Tao's An Introduction to Measure Theory: According to my viewpoint, the following are some salient features of the textbook:

  1. The text develops the theory from the bottom up. The textbook firsr covers the Jordan measure and the Lebesgue measure in $\mathbb{R}^d$, before moving on to cover measure theory on abstract spaces. This provides a concrete motivation for the subject without the loss of any rigor.

  2. Exercises are interspersed with the text. This is not a mandatory requirement, but I very much like that exercises are interspersed with the text. When and if one solve these exercises, it feels as if the person is actively working out part of the material in the textbook that hasn't been (explicitly) developed. Additional exercises based on examples, theorems etc. can be inserted in the textbook, but I think the textbook has a nice set of problems that accompany the development of the ideas in the textbook.

  3. Motivation, motivation, motivation! The textbook is rigorous, but it provides the necessary motivation and context of all the ideas developed. For example, here's a short excerpt from the section on Modes of Convergence:

If...one has a sequence $f_1, f_2, f_3,...$ of functions $f_n : X \longrightarrow \mathbb{R}$ or $f_n : X \longrightarrow \mathbb{C}$ on a common domain $X$, and a putative limit $f : X \longrightarrow \mathbb{R}$ or $f : X \longrightarrow \mathbb{C}$, there can now be many different ways in which the sequence $f_n$ may or may not converge to the limit f...This is contrast with the situation with scalars $x_n$ or $z_n$ (which corresponds to the case when $X$ is a single point) or vectors $v_n$ (which corresponds to the case when $X$ is a finite set such as $\{1, . . . , d\}$). Once $X$ becomes infinite, the functions $f_n$ acquire an infinite number of degrees of freedom, and this allows them to approach f in any number of inequivalent ways.

The last few sentences are well-written. They emphasize how the theory of convergence of sequences and vectors is implicit in the given problem, but, when $X$ becomes infinite, we're working on a function space that may admit nonequivalent notions of convergence w.r.t to nonequivalent induced norms. There are many other instances where I feel the textbook provides the necessary motivation and context.

It'd be great if someone else list textbooks in other areas that match at least some of the aforementioned criterion. I'm currently taking a second dig at measure theory, but I feel this is the clearest textbook written on the subject. It's concrete, in the sense that it starts from the ground up; it's completely rigorous; it provides necessary motivation and context in words; and the textbook is economically written. The textbooks by Folland, Royden, Rudin fail to meet at least one of the aforementioned criterion, at least in my opinion.

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    $\begingroup$ Did you check Cohn's measure theory? $\endgroup$
    – user370967
    Jan 10, 2019 at 8:57
  • $\begingroup$ @Math_QED No I haven't, but I'll look into it. I'm now more interested in looking at books in other fields that resemble Tao's book. $\endgroup$
    – user82261
    Jan 10, 2019 at 9:02

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For intro real analysis, the book "The real numbers and real analysis" by Ethan Block definitely fits all your criteria.

The book starts developping the real numbers, starting from the Peano axioms (and this takes like 200 pages) before diving in actual analysis content. Everything is proven, even details most textbooks don't even mention and it is very rigorous. Every concept is motivated, there are exercices after every section (no exercices at the end of an entire chapter) and history of the subject at the end of the chapter.

The book does assume familiarity with the mechanical side of calculus.

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    $\begingroup$ +1 a really good book. The first few chapters feel like Hardy's A Course of Pure Mathematics but more reader-friendly I'd say. $\endgroup$
    – BigbearZzz
    Jan 10, 2019 at 10:28
  • $\begingroup$ Thanks. I'm already familiar with this book, but I haven't gone through it. It'll be a good idea to skim this book some time. If you have any other suggestions, especially from other fields, please let us know. $\endgroup$
    – user82261
    Jan 10, 2019 at 11:14

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