Recently I have learned that at some math department the introductory course to Lebesgue integration not obligatory. Thus in another course on introduction to Hilbert spaces the $L^2(0,1)$ space is defined as the abstract completion of the space of continuous functions with respect to the norm defined by the Riemann integral of the square of a function.
I did not know that such an approach is possible. This motivated the question in the title. Particularly I would like to address it to algebraists and differential geometers, in other words whether the Riemann integral is sufficient for their professional purposes and the Lebesgue integral may be omitted from their background.