I would like to be able to define area, volume, distance ... in a unique way (as opposed to defining each term separately). One reason for that is that the geometric proofs of the Pythagorean theorem depend on the definition of area.

And intuitively, there seems to be a way: define n-dimensional measure of an n-dimensional region as "the number of points contained in the region".

For example, for area, since for natural numbers the product of 2 times 3 is equal to summing 3 twice, the area can be understood as summing the horizontal region (the number of points in a given segment) as many times as there are points in the vertical region. This process is consistent with thinking about an area as the "amount of painting"needed to cover the region, i.e. the amount of points needed to be painted, irrespective if the region is one dimensional, or two or three.

As a consequence, if we could think about our space not as the cartesian product of twice the real numbers, but a two dimensional lattice in a very fine mesh (smaller than the error measurement committed by our measuring device), we could define distance, area, volume ... in a unifying way: the n-dimensional measure of a region is simply the number of points inside the region.

But in order to define this is the Euclidean space, we should take the limit to a mesh tending to zero, and "completing" with all rational and irrational numbers (in the real line).

Can this process be done? If so, is there any book where the process is defined?