In some elementary analysis courses, we discussed what would fail without countable additivity, although it's not as if there would be some contradiction. It would merely be "not nice." We'd lose continuity under sequential monotone limits. Dunford and Schwartz say some things about finitely additive set functions. What happens when instead of notions like "$\sigma$ algebra" and countable additivity, you replace every occurrence of countability with some larger cardinal? So the reals may not be quite the right place to carry out such a theory because if you have an arbitrary sized disjoint union of "measurable" sets in the proposed theory, then unless all but countable many have "measure 0" we would have that the measure would have to be infinity or undefined in the signed case all the time. But what happens if more general values of the measures are taken, like say in a Banach space?

To the extent that the following question can be answered beyond "just because it worked out that way", why is it that our modern mathematical theories basically only care about scalar measures that are countably additive, or perhaps in strange situations banach space valued measures, but still countably additive? Why not finite, or why not bigger?

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    $\begingroup$ Actually this could be asked more generally, and it would start to get a bit philosophical. Metric spaces single out the countable cardinal too, as opposed to topological spaces where one uses nets. One could imagine structures of intermediate levels of generality, where to determine a structure you need to name the convergent nets founded on directed subsets of cardinality up to some upper bound. But such structures never seem to show up anywhere. Why, and what would happen if we tried a theory like this? A lot of things in analysis seem to stop at countable. $\endgroup$ – Jeff Oct 22 '13 at 6:45
  • $\begingroup$ I don't know a thing about the subject matter, but one idea might be to extend $\mathbb{R}$ with infinite and infinitesimal elements. Then maybe you'll find that the measures valued in this extended real number system (together with a maximum element $+\infty$) will need to satisfy a notion of additivity that is stronger than countable additivity. $\endgroup$ – goblin Oct 22 '13 at 6:45
  • $\begingroup$ By the way, this is an excellent question. Why is analysis so darn countable-centric? $\endgroup$ – goblin Oct 22 '13 at 6:46
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    $\begingroup$ @user18921: That's a whole other question, and its answer is quite simple. We start with finite, once we got a hold on that we moved to the next stage. Things that can be approximated by finite things, i.e. countable things. $\endgroup$ – Asaf Karagila Oct 22 '13 at 6:48
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    $\begingroup$ But it doesn't have to stop there right? $\endgroup$ – Jeff Oct 22 '13 at 6:48

I actually had the exact discussion with two friends when I was an undergrad and we took the measure theory course. Luckily for me, when we went to ask the professor teaching it, he was slightly familiar with model theory (his wife is a known model theorist) and he pointed out that you really just need some ordered abelian group structure to hold the measure. If you take some very very large group, you can have "measures" into that group instead of the real numbers.

All that is fine and dandy, but why don't we do it more often? The way I see it, there are two main reasons.

  1. Measure theory comes from problems that relate the the real world. In the real world we only do things over a finite course of time. However since we don't want to limit ourselves to a finite bound, allowing countable additivity allows us to go "as far away as we want, and then some" instead.

  2. Countable cardinalities are in some sense absolute. If you have two models of set theory with the same ordinals, they will always agree on the value of $\aleph_0$, but they may disagree on the value of $\aleph_1$. Moreover, even if they do agree on what is $\aleph_1$ they may disagree on whether or not $2^{\aleph_0}=\aleph_1$ or other, finer points.

    So once you leave the countable (or even separable) realm you will inevitably let set theory come into your house through the back door. Not a lot of people like that. For example Whitehead's problem is provable in the countable case, but is independent of $\sf ZFC$ in the uncountable case; or the fact that Calkin algebras can have outer automorphisms, or not depends on the continuum hypothesis.

So when you combine "countable additivity is good enough for what we needed measure theory for in the first place" and "we prefer to avoid elaborate set theoretical assumptions", you get the reason for why people are interested mainly in the countable/separable case.

In time, however, it stands to reason that people will find more and more interest in what happens beyond the realm of the countable, and when that happens people will either have to settle for some "canonical set theory" which decides a lot of statements (e.g. $V=L$ or so), or work with many different set theoretical assumptions. And frankly, to me, as a set theorist, neither sounds too appealing.

  • $\begingroup$ Can you elaborate a bit on the idea of replacing the real numbers with an Abelian group? A while back, I had the idea of replacing them with a partially ordered commutative monoid whose identity element is the least element, but never really pursued the idea. (Actually it was metric spaces that I was thinking about, but I'm guessing the issues that crop up are similar). $\endgroup$ – goblin Oct 22 '13 at 7:09
  • $\begingroup$ @user18921: I never really tried to write it down. But you can work these things yourself. It's just that when you define a measure, really all that you're using is the fact that there is an ordered group that you can map things into and compare which measure is larger. $\endgroup$ – Asaf Karagila Oct 22 '13 at 7:15

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