Mathematical rigor for engineers An engineer with a master's degree, I would like to improve my level of mathematical rigor. For example, I would like to come to grasp


*

*what measure theory has to do with the Fourier series and

*what group theory has to do with the eigenvalue,


among others.  I would like to understand why Lebesgue integration is important (the idea seems sort of trivial to me, which naturally means that I don't yet really get it), why the formality of Weierstrassian tests is preferable to the mere evasion of the problem by, say, the extraction of Hadamard's finite part, and so on.
To this end, I have been reading Georgi E. Shilov's Elementary Real and Complex Analysis, which I like; but at the rate Shilov is going it would take like 10,000 pages to advance as far as the Fourier series.  Therefore, I have looked instead on Amazon and Google Books for the book Mathematical Rigor for Engineers.
Unfortunately, no one seems to have written that book.
What are my options?
(To clarify: I do not mean to ask the opinion-based question, "What should I do?" Rather, I mean to ask the experience-based question, "What are my options?" Maybe the only answer is that what I seek resembles Financial Analysis for Kindergarteners or Transoceanic Logistics for Retail Clerks, but since I happen to possess an applicationist's acquaintance with mathematical productions like the Fourier Series and the eigenvalue, not to mention Hadamard's finite part, I thought that I would ask.)
 A: See Roman's 2-volume book [1] below. Although it's possibly the best reference I know for what you're asking (which has been asked fairly often over the years in various online math forums), his 2-volume book is almost never mentioned. The books [2] and [3] are rather well known and possibly others will mention them, the book [4] is fairly advanced but sufficiently reader-friendly to be worth looking at from time-to-time, and [5] is a bit less known (and a bit idiosyncratic).
[1] Paul Roman, Some Modern Mathematics for Physicists and Other Outsiders, Volume 1 (1975): An Introduction to Algebra, Topology, and Functional Analysis (Volume 1 contents) AND Volume 2 (1975): Functional Analysis with Applications (Volume 2 contents -- click on amazon.com's "look inside" for Volume 1; Volume 2 contents are on pp. x-xi)
Review in Physics Today
Volume 30 #5 (May 1977), pp. 72 & 74; review by Andrew Lenard (1927-2020)
Review in Computers and Mathematics with Applications
Volume 3 #1 (1977), pp. 83-84; review by  Wilhelm Ornstein (1905-2002)
[2] George F. Simmons, Introduction to Topology and Modern Analysis (1963)
[3] Thomas A. Garrity and Lori Pedersen, All the Mathematics You Missed: But Need to Know for Graduate School (2001)
[4] Charalambos D. Aliprantis and Kim C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide (2006)
[5] Robert Hermann, Lectures in Mathematical Physics Volume 1 (1970) and Volume 2 (1970)
A: I think that what you're really looking for is Representation Theory. Now the problem is that it can be quite advanced for an engineer trying to approach this subject from a mathematical point of view. But maybe it can be done from a more "physical" point of view. 
I would suggest to you "Lectures on Linear Algebra" from Gelfand that if I rember correctly also adresses Fourier Series, and then the second volume of "Principles of Advanced Mathematical Physics Volume 2" of Richtmeyer which has in the first 10 chapter the basis of representation theory in a very comprehensible way. I'm not sure on this last one since it might be too advanced, but I would give it a try if I were you.
A: For the Fourier series<->measure theory connection, you can perform some operations using infinite series as a black box but in order to describe a Fourier series corresponding to a discontinuous function you will (eventually) begin to move into measure theory. Please remind me to respond after this semester is over, as I am enrolling in a course on this exact question which starts next week (my undergrad didn't emphasize the "applied" side outside of the Physics department).
Group theory is connected to eigenvalues because the square matrices form a group and eigenvalues allow you to detect properties of an individual matrix (such as whether it can be decomposed in any of several ways, what the "long-term behavior" of a Markov chain will be, whether the matrix will be stable or unstable under exponentiation even if it lacks the Markov property, etc.) I would be happy to answer further questions on this subject next week.
A: I think it's difficult to beat "Real Analysis" by John. M. Howie from Springer. I come from an engineering background and only started reading this book after I had completed my undergrad & MSc in engineering: having realized that after 5 years of studying, I still completely lacked the rigorous basics of Mathematics.
The book above is "thin" enough to be read within within 1 year including exercises (I had gone through the entire book just using my weekends: took me less than 12 months). It most definitely provided the basics that then allowed me to progress further.
Some of the topics you mention (i.e. Rigorous Lebesgue Integration) are at what I would describe "Graduate-level Mathematics": the book suggested above is for udergrads. But the approach I took is to start with that and then selectively study graduate topics that were of interest to me (i.e. Lebesgue integrals, probability measure theory).
I think the above approach is better than just picking up a thick book with Graduate-level mathematics, because as you point out: it would take too much time to try to build up "general graduate-level mathematical knowledge" by self-learning.
After reading the book from Howie, I then for example went through the MIT open-courseware lecture notes here: I felt the notes start where the book from Howie "ends", i.e. the book allowed me to "pick up" Grad-level topics that would have been previously unapproachable to me.
PS: It is an interesting topic in itself why Engineering courses (including graduate-level) do not teach mathematics rigorously. I think it's a huge problem and should be addressed.
