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I am starting my $2^{nd}$ undergrad analysis class in which we will cover chapters 1-4 of Stein's book (includes Measure Theory, Lebesgue Integration, Dominated Convergence Theorem, Hilbert Spaces, ...).

I have looked at the textbook and it looks traditional; this is good, but I am looking for a secondary book which is more informal/visual/intuitive which will help me understand the material better and build more intuition.

The following are examples of books from other fields which are comparable to what I'm looking for now: Visual Complex Analysis by Tristan Needham for Complex Analysis, Understanding Analysis by Stephen Abbott for basic Real Analysis, and The Art and Craft of Problem Solving by Paul Zeitz.

Thank you.

Edit: One explicit characteristic that I'm looking for is that the book gives details on how one might have come up with a proof, not just gives the most polished version of the proof.

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In case you speak German, (ovi is the national drink from switzerland), here is an excellent Script:

https://people.math.ethz.ch/~struwe/Skripten/Analysis-I-II-final-6-9-2012.pdf

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    $\begingroup$ Ah unfortunately Ovi is a Romanian name :( but thank you +1 anyway $\endgroup$ – Ovi Jan 27 '18 at 22:01
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You might be interested in checking out A Radical Approach to Lebesgue's Theory of Integration by David Bressoud.

Here's the description from Amazon:

This lively introduction to measure theory and Lebesgue integration is motivated by the historical questions that led to its development. The author stresses the original purpose of the definitions and theorems, highlighting the difficulties mathematicians encountered as these ideas were refined. The story begins with Riemann's definition of the integral, and then follows the efforts of those who wrestled with the difficulties inherent in it, until Lebesgue finally broke with Riemann's definition. With his new way of understanding integration, Lebesgue opened the door to fresh and productive approaches to the previously intractable problems of analysis.

Sheldon Axler, whose textbook Linear Algebra Done Right is very popular, is writing a book on measure theory. A chapter is already available on his website.

I also recommend The Calculus Gallery: Masterpieces from Newton to Lebesgue by William Dunham.

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    $\begingroup$ At a higher level than Bressoud's book (and perhaps not quite as chatty as Bressoud's book, but still full of interesting comments and sidelights) is Bruckner/Bruckner/Thomson's Real Analysis, which is freely available on the internet. $\endgroup$ – Dave L. Renfro Jan 30 '18 at 20:51
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This link give you free acsses to MIT courses and this gives you access to videos tools along with the following with multiple references on calculus and real analysis for beginners Calculus book recommendations (for complete beginner) and Good First Course in real analysis book for self study

I have used this book: at the second hand of my starting point in such area

I was recommended to use this book on: Lebesgue's Theory of Integration but I never had access to it may be good enough.

I also think this is a fairly Good lecture notes

Our professor uses A Friendly Introduction To Analysis. I think it reads fairly easy.

You should also try "Foundations of Modern Analysis" by J. Dieudonne. you will be learning from true Master. In his book Dieudonne do not assume any previus knowledge about analysis or even mathematics at all.

also see this Online Course for Real Analysis and Video Lectures that closely follow Rudin's Real Analysis/Royden's Real Analysis

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My favourite all-rounder is Lieb and Loss "Analysis" https://bookstore.ams.org/gsm-14-r the authors manage to combine a very pedagogical tone with relatively advanced topics.

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Introduction to Real Analysis
Introductory Real Analysis

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