Various "sizes" of 0-measured sets

Question

I am looking for a formalization of an intuitive concept of size, in cases simple measure is too coarse. It will be easier for me to give an example.

Let $\mu$ be the Lebesgue measure on the unit square $I^2$. Consider the following sets, all of measure 0:

• $A_1=\left\{\left(\frac{1}{2},\frac{1}{2}\right)\right\}$
• $A_2=I^2\cap\left(\left\{\frac{1}{2}\right\}\times\mathbb{Q}\right)$
• $A_3=I^2\cap\left(\mathbb{Q}\times\mathbb{Q}\right)$
• $A_4=I^2\cap\left(\left\{\frac{1}{2}\right\}\times\mathbb{R}\right)$
• $A_5=I^2\cap\left(\mathbb{Q}\times\mathbb{R}\right)$

If $\sigma$ is some formalization of my intuition of size in these cases, I would expect $$\sigma(A_1)<\sigma(A_2)<\sigma(A_3)<\sigma(A_4)<\sigma(A_5).$$

I know that $A_1,A_2,A_3$ may be distinguished from $A_4,A_5$ according to their cardinality, but that's again too coarse.

Another issue is this: what sort of functions will be $\sigma$-preserving? Of course, bijections will not suffice, as they may take $A_4$ into $I^2$, which is naturally "larger".

I believe there should be some term or theory which investigates finer "sizes". Do you know of such?

Answer 1

Look at the concept of Hausdorff dimension! Interesting!

Answer 2

I collected together a number of different concepts here a few years back:

https://en.wikipedia.org/wiki/User:CRGreathouse/Large_and_small_sets

They mostly focus on the discrete, though.