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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).
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I'm a bit confused by this question, as not all measures are real. See – A Walker Sep 21 '12 at 15:18
@AWalker: Hah, I didn't know about them. I still think there's a legitimate question here, though. – Ben Millwood Sep 21 '12 at 15:20
In the study of higher dimensional local fields one encounters the problem that from dimension 2 on they are not locally compact anymore, so there is no Haar measure on the additive group of the field. There are several attempts of circumventing the problem by using measures that take values in for example $\mathbb{R}((X))$. See – Michalis Sep 21 '12 at 15:26
The fact that measures of sets can be multiplied and divided is essential in probability theory; it's the basis for the absolutely fundamental notions of conditional probability and independence. – Nate Eldredge Sep 21 '12 at 15:28
@John, There are $p$-adic measures (i.e. $p$-adic valued), but they are not countably additive. They are finitely additive functions valued in, e.g., the ring of integers of $\mathbf{C}_p$, defined on the set of compact open subsets of some profinite group $G$. They correspond to elements of certain "completed group algebras," a typical example being $\mathbf{Z}_p[[G]]$, where $G$ is any profinite group. This is, by definition, the inverse limit over the group rings $\mathbf{Z}_p[G/U]$, where $U$ ranges over all open normal subgroups of $G$. – Keenan Kidwell Sep 21 '12 at 16:20
up vote 2 down vote accepted

The purpose was to define integrals in such a way that it is easy to prove that in some circumstances $\displaystyle\lim\limits_{n\to\infty} \int_A f_n(x)\,dx = \int_A \lim\limits_{n\to\infty} f_n(x)\,dx$ and things like that. If there are some other purposes that it doesn't serve, that's not a criticism of it.

You cannot abandon the Archimedean property and keep completeness unless you throw away some of the algebra. In a non-Archimedean ordered field, the set $\{1,1+1,1+1+1,\ldots\}$ is bounded above. With completeness, it would have a least upper bound $c$. Then $c-1$ would not be an upper bound, so $c-1<\underbrace{1+\cdots+1}_{ \begin{smallmatrix} \text{some finite} \\ \text{number of} \\ \text{terms} \end{smallmatrix} }$, so $c<\underbrace{1+\cdots+1}_{ \begin{smallmatrix} \text{just one} \\ \text{more term} \end{smallmatrix} }$, but then $c$ is not an upper bound of that set.

As for countable sets having measure $0$, notice that sets of size $1$ have measure $0$, and measure is countably additive, so of course countable sets must have measure $0$.

Can one define some measure-like functions on sets of reals that would have some list of properties that you like? Probably one can.

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There are two approaches to measure and integration. One starts with measure theory and derives integrals from measures, the other develops integrals on their own and treats measures as integrals of indicator functions.

Now integration makes sense in a lot of structures. It is essentially a generalization of "taking averages" and averages are well defined for vector spaces. So there is a rather rich theory of integration in vector spaces (with additional structure) and a corresponding theory of vector measures.

Now, whether a specific mathematical formalism fits well to a certain problem is of course always subject to (extramathematical) debate. Certainly, the "special case" of real valued measures has proven to be both rich enough to give rise to a lot of interesting mathematics and has found many applications, most notably probability theory.

The question whether real numbers provide the appropriate notion of "size" is a separate question. There are axiomatic approaches to measuring the size of objects in the mathematical theory of measurement and the first such results, due to Helmhlotz an Hölder, were to the effect that quantities can be well-represented by real numbers.

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