> I don't like this attitude at all. Euclid, Archimedes, Newton, Leibniz, Riemann, and many others did a fine job correctly calculating areas, volumes, and so on without need of measure theory.

The history of mathematics is a history of ever-increasing standards for proofs. "Proofs" that, say, Euler wrote down that were perfectly valid for the standards of their time would probably not accepted in todays standard. I personally also think it's plausible that today's proof standard in "typical" math papers will not be acceptable in 100 years anymore, when they perhaps will additionally have to be computer-checkable (and computer-checked).