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I often introduce the Lebesgue integral (say, for positive functions) as a direct generalization of the Riemann/Riemann-Darboux one just saying that the only differences are that we now use measurable sets instead of intervals as partition elements and take countable partitions instead of the finite ones. This is not how it is written in most textbooks but this allows me to capitalize on the knowledge of the Riemann integration and emphasize both similarities and differences in a rather neat way.

The proofs of some of the basic results in measure theory are intricate but also rather boring, and I wonder whether any analysts will stand up to say that they learned something from them and use them in their work.

I will. If it were just for the two covering lemmas alone (Vitali and Besicovich), I would already vote to have the measure theory courses running. Frostman's lemma, Hausdorff dimension, Sarde's theorem (the full version, not the baby one, of course) jump to the head next. Area and Coarea formulae for Lipschitz mappings form the next layer. And so on. As to proofs, the fact that the least monotone classes containing a ring is a sigma-algebra is shocking enough to wake a student up (and the proof is 3 lines). The dreaded Fubini follows from it in 6 more lines once the monotone convergence results are already in place if you do not bother to consider sigma-finite case rather than just finite or pass to the completion of the measure (and those should really be given as exercises).

It is true that you can make the course boring quite easily if you choose to. The surest way to do it is the same as for an English course: instead of reading poetry, spend all the time perfecting the knowledge of the alphabet.