Well, maybe I will not be mathematically absolutely strict (or even in some aspects correct), but still in some textbooks I read the following line of reasoning. First the authors start with the derivation of Fourier series for periodic signals (I am speaking about book, dealing with physical processes). Second they come to nonperiodic signals, defined (lets say for easiness) on some interval and periodically (with the period, equal to that interval) complement it to form a periodic one. Then they expand the obtained "periodic" signal in Fourier series and at last they take the limit with the period approaching infinity (to get in the limit an initial nonperiodic signal). And in the limiting form the series representation turns into integral one.