Added later:My hat is off to Harry for this question, and all who have answered. I learned a lot from them, especially short ones like Allen Knutson’s. Building on them, I throw out a possible combined approach. (It helps if the student read Euler’s precalculus book with infinite series but never mind.)
I address those beginning students who care about understanding what is done and why, more than seeing it the absolute “best” way on first encounter. I.e. I suggest the first job is to motivate the best way, not present it in full.
The basic problem is to define a way of adding up, or averaging, values of a function with an uncountable number of values. All competing definitions seem to use limits of functions with a finite set of values, or “simple functions”. Thus the only question is which simple functions to use and what method of taking a limit.
One could begin with the integral of a monotone continuous function, as did Newton, and exhibit the two standard ways of approximating it by integrals of simple functions, namely adding up over columns (Riemann), and adding up over rows (Lebesgue). Here they are almost the same, in that one representative value is chosen for all points in a subinterval of the domain. From a computational viewpoint, Riemann’s method is superior since the length of the domain subintervals is more easily calculated. Thus for elementary approximations, Riemann’s method is worth knowing.
Introducing more complicated functions one observes how the domain “level sets” become more complicated in Lebesgue’s approach, making the definition of their size a significant task, while Riemann’s approximations remain easy to calculate.One could then define the Riemann integral and state its fundamental theorem, and define “negligible sets” and state the Riemann-Lebesgue criterion for Riemann integrability, or just mention the special cases of continuous and piecewise monotone functions, possibly proving the latter, or the former assuming uniform continuity. One could also state and/or prove the convergence theorem involving uniform limits, possibly informally.The characteristic function of the rationals, whose integral should exist for anyone knowing infinite series, but does not in Riemann’s sense, shows one limitation of this approach, whetting the appetite for Lebesgue. It also explains the failure of convergence for pointwise limits.
One might admit that much of the work in the first course will use the technique of antiderivatives rather than integrals, accepting in spirit Dieudonne’s argument, but that Riemann sums are crucial for approximations. For the technique of evaluating integrals by antiderivatives, one can observe the usual proof of the FTC works on continuous functions for any definition of the integral which is monotone and additive over subdivisions of the domain.
Point out it is hard to give a full fundamental theorem for Riemann integration, i.e. to intrinsically characterize indefinite integrals of all Riemann integrable functions. One could mention the advantages of Lebesgue’s definition.
In this context one could discuss a more flexible definition of antiderivative as Dieudonne’ does, at least for step functions, and remark that one can characterize indefinite integrals of bounded functions with only a finite (or countable) set of discontinuities.
Since many instructors in the US omit proofs entirely in beginning calculus, the question of whether to present complete proofs for Riemann integration applies more to a beginning analysis course, where I think it can prove useful practice for many students.
To the argument that presenting the most advanced and abstract version first is more efficient, I would point out that having been taught that way, it took me another 40 years to understand what is going on, hence not very efficient for me.