To measure or not to measure... a real problem!
Pt En This post will be more theoretical than usual. If you are afraid of mathematics, go away now before it is too late! I will write about a set, the Vitali set, which arises in measure theory , to show that there is no coherent way of assigning a size to every subset of the real line. Because of that I will try to define size for subsets of the real line and then verify that such task is impossible. I will also enforce a couple of (seemingly reasonable) restrictions on my definition. I challenge you to read this post and to calmly follow my reasoning. Whenever something doesn't seem obvious, try to make it clear by yourself with a piece of paper and pen/pencil. If any doubts persist, drop me your question(s) in the comments and I will answer gladly! Let us call $m$ to the function that, given a subset of the real line, returns its size; that is, let us try to define $m: \mathcal{P}(\mathbb{R}) \to [0, \infty]$. Let us also suppose that the function $m$ sat...