As the title indicated, I am wondering if one (probably as an undergraduate math major) can learn much of Fourier Analysis, without taking a course in integration theory.
I am taking a very light introductory course in Hilbert space (using N. Young as the textbook), which does not need any integration theory as pre-requisite. Oftentimes I saw something like "the function behaves like this almost everywhere and hence [some properties]". So please perhaps answer this question:does one really need integration theory for Hilbert space theory, and further functional analysis?
Also for a side note, it gives me the impression that my engineer friends (EE major as such) can deal with Fourier transform/convolution/etc rather comfortably.