Measures, as I understand it, exist to give us some sense of the "size" of a set. However, there's a lot of detail they gloss over, sometimes – the usual measure ignores all countable sets, for example. What I'm wondering is if this is not because the real number it gives us is in some sense not "rich" enough to reflect that information.
I understand that the real numbers are the unique complete totally ordered Archimedean field. I can see that:
- Totally ordered seems pretty essential. Indeed, a definition of "size" that wasn't comparable would be fairly nonsensical.
- Complete seems sensible, given how many theorems in measure theory are based around convergence.
- Field... well, measures often actually take values in $[0,\infty]$, rather than $\mathbb R$, so it's not a field. Certainly I can see that having addition form a commutative and injective monoid (injective, that is, apart from $\infty$, which might cause problems) makes sense. Multiplication and division, as a commenter has pointed out, are key in probability theory, particularly in the definition of conditional probability, so I can see how they are desirable.
But the Archimedean property is where I start to be skeptical. As I see it, the size of sets isn't intuitively an Archimedean structure. No matter how many small sets I have, I can't make a bigger set, for any of several reasonable definitions of "small" and "big".
So, is it possible that there could be some other structure, possibly non-Archimedean, or not a field, that could serve as the codomain of a measure, and provide richer information about the "size" of the set? Or is there something inevitable about the real numbers in this context that I'm not seeing?
To pre-empt some possible answers:
- I am willing to accept that measures evolved as real-valued functions out of convenience, but I would be surprised if no-one ever thought of doing it differently. So I think any answer that does not at least consider the possibility of alternatives isn't what I'm looking for.
- I realise that "as much information as possible" isn't necessarily the aim of a measure, and it gains some expedience by abstracting away from sets we think too small to care about. But just as sometimes you want a coarse tool (say, "empty or non-empty"), that throws away boring detail, sometimes you'd like a finer tool (cardinality or volume or...) that abstracts less.
- A commenter has pointed out to me that there exist complex- and vector-valued measures, which I did not know. This is interesting, but doesn't seem to give richer size information, rather modelling another phenomenon entirely (e.g. signed measures can be used to model charge, I am told). So that's probably not what I'm looking for. (I notice that you can define a vector-valued measure by just taking a set to its indicator function, losing zero information. But then measures aren't comparable, so I don't consider this a reasonable notion of volume).