I am a freshman electrical engineering student and I just finished calc 3 and vector calculus. During the semester, throughout the lectures (during the Integral part of calc 3 and vector calc) our professor would say things like "Jordan measure" or "Lebesgue measure" and other things that I don't remember off the top of my head, so one day I went and asked him what exactly he meant with these. He said there is a whole branch of mathematics called Measure Theory and one of its core purposes is to help us improve our techniques of integration. He also told me that the reason that engineers are only taught up to triple integrals and no higher (for example we don't learn quadruple integrals, and I couldn't find anything online about them either) is because the Riemann integral is hard to generalise to higher dimensions and we would have to use a different kind of integral, called the Lebesgue integral for that purpose, which unfortunately engineers typically don't learn. He also told me that this type of integral is also widely used in Fourier Analysis, which I am about to take next semester.
All this really got my attention and I am really curious to learn more about Measure Theory, so I am planning to buy a book. Can you suggest a book that is beginner friendly (I mean complete beginner) and also intuitive? One that shows why we need measure theory and how it arose?
I am also open to general advice regarding this endeavour (where I should start, how I should continue) as I have heard it's not the most intuitive subject to learn and it takes patience and dedication.
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