# Gracious Living

Measures and Non-Measurable Sets
November 23, 2010, 14:22
Filed under: Analysis, Math | Tags: , , , ,

This is very late, but don’t worry, I’ll get another one up tonight.

One of the big lessons learned from the Banach-Tarski paradox is that even in something as simple as a unit ball, we can find sets of impossible or undefinable volume. In the discussion preceding the proof, I also mentioned the paradoxes surrounding length of fractal curves in $\mathbb{R}^2$ and area in $\mathbb{R}^3$. Together, these present us with a crisis: how can we characterize length, area, and volume? The answer to this crisis was developed around the turn of the century by heroes like Borel and Lebesgue, and it’s called measure theory.