In any standard development of measure theory in several well-known textbooks, the use of almost everywhere (a.e.) defined functions are first seen in the statement of Fubini's theorem, which states, in particular (under standard notation) that
$$\int f(x, y) d\mu(x) = g(y)$$
is defined almost everywhere. Now $g$ becomes an a.e. defined function. It is indeed true that the development carried out so far in the book can be formulated in terms of a.e. functions, and not lose much, but still, I think it deserves a remark, if not a full-blown methodology that uses only a.e. functions right from the beginning. This latter suggestion is justifiably not adopted, because it gets in the way of communicating the main ideas.
However, most sources I've seen completely avoids even mentioning this fact (one exception I've seen is Tao's Analysis II). Why is this the case?
P.S.: I've also not seen any advanced book seriously developing the whole theory consistently using a.e. defined functions (one exception is Fremlin's monograph). Why is this stuff left unformalized? Is it just because its a notational hassle, or is there some proper reason?