@Tim. Physical chemistry, organic chemistry, inorganic chemistry and so on. I suppose quantum chemistry means use of quantum mechanical methods to understand molecules and their interactions with each other, structures of crystals and other states of matter, and in short everything in chemistry that has need for quantum physics. I think if chemists do the work, they call it quantum chemistry, instead of calling it quantum physics.

Ah thanks a lot for the reference and explanations! You have a soft corner for Chemistry?

That is the book. The first part is written for Serre's wife, Josiane Serre, for teaching her students in quantum chemistry. Somehow it is all very relevant there, though I have no idea. Maybe someone else can explain it better, with references to Chemistry books.

Maybe you or Qiaochu should write that down as an answer?

Oh that is good to know. I always wondered why. Thanks for pointing out.

@Ryan, I believe that it is weyl, not Weil.

It gives a good notion of "almost-everywhere", and it is psychologically much more satisfactory for me that way.

However your point of view is extremely fruitful, as shown by the great success of noncommutative geometry.

I think the viewpoint taken by you, ie represent a compact Hausdorff space by its $C^*$-algebra, and declare a measure to be a functional on it(like in Riesz Representation theorem) is what is adopted by noncommutative geometers, for their noncommutative analogies. So it is not true that analysts are not willing to take up category theory. Still, the base of the subject is in good old Lebesgue theory, and that must be first carried out in the traditional fashion. That is what analysis is all about. The essence of analysis is contained in that type of stuff, not in category theory.

This is not an answer. But perhaps looking into the "Eisenstein ideal" paper of Mazur might help. There he was constructing some weird quotients. The Eisensten quotient seemed to be not the smallest one, but not the largest one either. I didn't get into all the intricacies since I am not an expert. Perhaps thinking about such quotients might shed more light on your conjecture.

I have also argued that many theorems make use of the notion of almost everywhere, and also that this notion becomes much neater in the presence of a complete measure. Both theorems are instances. Wikipedia explains some ugliness in the case of Fubini. Of course, you can do measures without completeness, as for instance the Riesz representation theorem uses only Borel measures, and on the other hand you can see also every Borel measure as a functional. That's not the issue. The point is that Lebesgue theory is neater because of completeness.