The following example was brought up in two different classes that I am taking, within a couple days of each other. It is the classic example of an immeasurable set on the interval [0,1]. Now in math, words like measure and measurable have technical definitions, and lead to bizarre results like the Banach-Tarski paradox. I’m going to give a fairly informal explanation here. Continue reading
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is a fractal that is obtained by repeatedly removing the middle third of a segment. Start with the closed interval ![[0,1]](https://s0.wp.com/latex.php?latex=%5B0%2C1%5D&bg=ffffff&fg=323232&s=0 "[0,1]")
. Remove the open interval 
to obtain ![[0,\frac{1}{3}] \cup [\frac{2}{3},1]](https://s0.wp.com/latex.php?latex=%5B0%2C%5Cfrac%7B1%7D%7B3%7D%5D+%5Ccup+%5B%5Cfrac%7B2%7D%7B3%7D%2C1%5D&bg=ffffff&fg=323232&s=0 "[0,\frac{1}{3}] \cup [\frac{2}{3},1]")
, i.e. two disjoint closed segments. Remove the middle thirds of those two segments, and you end up with four disjoint segments. After infinitely many steps, the result is called the Cantor set. The diagram below is only an approximation after a finite number of steps.
