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?

  • 4
    $\begingroup$ Most elementary analysis books do not use measure theory. Most such books are directed primarily at scientists and engineers, who have, in general, very little if any use for measure theory. $\endgroup$
    – Dan Fox
    Commented Feb 5, 2020 at 13:25
  • $\begingroup$ @DanFox: I understand. I was referring to books on elementary measure theory. $\endgroup$ Commented Feb 5, 2020 at 13:37
  • 2
    $\begingroup$ I'm surprised by this - I remember when learning analysis that the professors I had were very direct in defining $L^1$ to be a space of a.e. equivalence classes of functions and using this definition whenever something like Fubini's theorem came up - of course, every book proceeded to act as if these classes were just functions, since it rarely mattered. I don't recall what textbook was used, though. $\endgroup$ Commented Feb 8, 2020 at 3:42


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.