How can we even measure something?

No really, think about it. Measuring stuff is kinda weird…

To start off, think about measuring area… pretty normal stuff… let’s say you are given 2 circles, one with a radius of 1m and another with a radius of 2m.
Instinctively, you know the areas of these circles: $\pi m^2$ & $4\pi m^2$. All of this is fine and all, but at its core, what is a circle? “A region” or “a set” in 2D space. So let’s consider circle A to be the 1m one, and circle B to be the 2m one. So,

$$A = \{(x,y) \mid x^2 + y^2 < 1\}$$

$$B = \{(x,y) \mid x^2 + y^2 < 4\}$$

I hope you’re convinced that the circles can be represented by these sets, because the next bit of logical reasoning kinda depends on it.
So, now let’s start with the main problem here. Did you know that you can map each element of B to one element of A? No seriously… pick one element from B, let’s say $(b_x, b_y)$ — just divide both elements to get $(b_x/2, b_y/2)$. So for each element of B, there exists an element in A. Technically, that should mean the size of set B is equal to A… which means… yeah, no — that doesn’t really make sense, right? Set B is clearly bigger… so measurement isn’t clearly just about the set. Although it talks about a set, what does it even mean then? (Example of this map is here)

See, this is the hook that always intrigued me about measure theory. The only thing human beings are good at is set theory, so essentially everything is defined as sets. So where does our general idea of measurement fall under that umbrella? And this isn’t even the original thing that bothered me…

The original problem was something like this:

Say we have a uniform probability distribution. In normal English: let’s say we can have any number from 0 to 1, and all of those numbers have equal probability of getting picked — that’s essentially what it means to have a uniform distribution. So if I say, “What is the probability of picking 0.1 specifically?” — I can’t say the probability is 0, because picking 0.1 is a very possible thing (though a very unlikely one). The best we can do is pick a range, for which we can calculate the probability. Why is this the case? Why can’t we do it for a single event / single number?

Same thing goes with calculus and limits. A very infamous example exists where you can wrap a $2m^2$ square around a 1m radius circle, with no change in area, which clearly violates our intuitive notion of reality. They should not have equal areas. So why does that limit not work, when all the other limits clearly do work — or else nothing would come out of calculus? What’s the difference?

This is my journey of discovering why some measurements can be made, why some can’t. And also, what exactly does it mean to measure something?