There are many good arguments, that one should be properly trained in the use of Lebesgue's integration theory before starting to work in Fourier Analysis. More generally, measure theory combined with the existence the Haar measure over any LCA (locally compact Abelian group) is taken as the basic ingredient for Haarmonic Analysis. In fact, Lebesgue integration theory necessary to define both convolutions or the Fourier transform through integrals, and thus the use of the space L1 of Lebesgue integrable function is often the first steps towards Fourier analysis, despite all the shortcomings, such as problems with the Fourier inversion theory or the verification of Plancherel's theorem for L2.

In this talk I will indicate how a purely functional analytic approach, simply using basic Banach space theory, can be used to go a long way from translation invariant system, convolution, the Fourier transform etc.. In principle I will disclose provide a motivation for some of the material usually done in my more recent Fourier analysis courses at the University of Vienna. The far goal being (beyond this lecture) to motivate the description of all these operations in the context of generalized functions, more concretely, using the Banach Gelfand Triple based on the Segal Algebra S_0(G).