Various "sizes" of 0-measured sets 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?

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.
