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Basically, a lebesgue measure of dimension $n$ of a set of the same dimension $n$ is $n$-volume, $\lambda_n(S)$.

If the dimension of a set is greater than the dimension of the measure, the measure is considered infinite: a length of a square is infinite, just as area of a cube.

But did anyone make attempts not to just consider it "infinite" but make it more refined, by assigning to such measures infinite but precise values?

For instance, the number of points in a set is $0$-volume. It is a Lebesgue measure of order $0$. Such measure of a set of dimension $1$ will be infinite. But intuitively, $\lambda_0{(0,1)}<\lambda_0{[0,1]}<\lambda_0{(0,2)}<\lambda_0(\text{unit square})$

In other words, the "number of points" of a bigger interval or a set of greater dimension is bigger.

I feel that this may be a basis to introduce infinite quantities, even corresponding to uncountable infinities.

In my impression, this would go beyound, say, surreal numbers.

I wonder whether such idea was ever investigated?

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    $\begingroup$ There is at least one way to organize all these infinities of different dimensions into meaningful quantities. See e. g. What is the length of a potato? by Schanuel $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Apr 23, 2023 at 18:02
  • $\begingroup$ @მამუკაჯიბლაძე yes, this is exactly what I was asking about. Still reading. $\endgroup$
    – Anixx
    Commented Apr 23, 2023 at 22:35
  • $\begingroup$ @მამუკაჯიბლაძე well, so Euler's characteristic is finite (regularized) part of numerocity... $\endgroup$
    – Anixx
    Commented Apr 23, 2023 at 22:54
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    $\begingroup$ @მამუკაჯიბლაძე the linked article says "Of these two advances, Euler's has been by far the more important; but we seem, most of us, to have spent more effort retraining our intuitions to incorporate Cantor's ideas than Euler's." $\endgroup$
    – Anixx
    Commented Apr 23, 2023 at 23:27
  • $\begingroup$ @მამუკაჯიბლაძე also, this seems to be something similar en.wikipedia.org/wiki/Motive_(algebraic_geometry) $\endgroup$
    – Anixx
    Commented May 24, 2023 at 1:24

1 Answer 1

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Such an approach will violate the Cantor-Hume principle, which asserts that "the number of elements" of a set $A$ should be invariant under equinumorsity. That is, if $A$ and $B$ can be placed into one-to-one correspondence, then they should have the same number of elements. This principle is often defended as a fundamental principle for any concept of "the number of".

If one combines this also with the idea that if $A\subseteq B$, then the number of elements of $A$ is $\leq$ the number of elements in $B$, it follows that we cannot commit to the "intuitive" strict inequalities you mention for (0,1), [0,1], (0,2).

Those intuitive strict inequalities are hugely discussed (hundred of papers) in the philosophical literature surrounding Hume's principle, with the main point being the tension between that principle and Euclid's principle, asserting that the whole is (strictly) greater than the (proper) parts. Galileo famously pointed out this tension when he considered the perfect squares as a part of the natural numbers. On the one hand, they are in one-to-one correspondence, so they should have the same number of elements; on the other hand, most numbers are not perfect squares, and so the number of numbers seems bigger than the number of perfect squares.

My main point is that to have a solution of the kind you seek, your number-assignments will have to violate the Cantor-Hume principle, which will mean for many people that you are not speaking of "the number of" any longer.

And therefore you should probably itemize exactly which features you want to have in your measures. Finite additivity? $\sigma$-additivity? Achieving this with any generality will be difficult in the surreals, which are not order complete.

Meanwhile, one can consider various nonstandard counting measures, which provide a nonstandard measure similar to what you mention. Basically, one takes a nonstandard finite set $I$ in the nonstandard reals $\mathbb{R}^*$, and then for any set $A$ you count the size of $A^*\cap I$ as a nonstandard finite set.

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  • $\begingroup$ Thanks for pointing out at the Hume principle, but my question is not about cardinality but rather about numerocity (Lebesque measure of dimension $0$) or, about generalization of Lebesgue measures in general. The concept of numerocity is discussed for instance, here: mathenchant.wordpress.com/2018/09/16/a-new-game-with-infinity The numerocities of non-dense subsets of reals are quite easy: they can be equated to divergent series, thus to Hardy fields elements or by surreal numbers. The mistery appears when we want to represent numerocities of dense sets, and especially of continuum. $\endgroup$
    – Anixx
    Commented Apr 23, 2023 at 22:11
  • $\begingroup$ "which will mean for many people that you are not speaking of "the number of" any longer" - this is pity that people equate cantorian cardinality to the "number of elements". $\endgroup$
    – Anixx
    Commented Apr 23, 2023 at 22:13
  • $\begingroup$ "Meanwhile, one can consider various nonstandard counting measures, which provide a nonstandard measure similar to what you mention." - this is exactly what I was interested in, because it seems to me that measure theory may be powerful enough to handle non-countable numerocities. $\endgroup$
    – Anixx
    Commented Apr 23, 2023 at 22:15
  • $\begingroup$ If we talk about the non-dense subsets of reals, we can equate the numerocity of the set of roots of a function $f(x)$ the following numerocity: $\int_{-\infty}^\infty \delta(f(x))|f'(x)| dx$. This means, we even can write down the expression for numerocity of any subset of reals as an integral, but the exact meaning of such integral would need clarification. $\endgroup$
    – Anixx
    Commented Apr 23, 2023 at 22:28
  • $\begingroup$ You call it a pity, but to me it appears to be an extremely illuminating and enormously successful theory. Numerosity, of course, is simply a bit of vocabulary meant to suggest "number", but the fact remains that it violates the Cantor-Hume principle. Galileo had thrown up his hands in confusion at the conflict between Euclid's principle and Cantor-Hume, opting ultimately to keep Euclid. It seems numerosity is motivated perhaps by the same idea. $\endgroup$
    – Joel David Hamkins
    Commented Apr 23, 2023 at 22:38

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