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.)