The Measure of a Ball
I’ve been looking into measure theory lately. Like quantum physics, measure theory is one of those things that underpins absolutely everything in statistics, and yet surprisingly rarely has an impact on what we’re doing when we’re using statistics. Hence why I had never really come across it until I started looking into Bayesian nonparametrics. Now I know how physics students must feel when they first start to learn about sub-atomic particles. Suddenly, measure theory is everywhere.
Of course, just as there is a Brian Cox for quantum physics, there is Terence Tao for measure theory. He has literally written the book on it, and that’s the book I’m using to learn more about it. Today, I want to discuss a simple example from the start of the book, and how it helped me better understand elementary measures.
The fundamental question at the heart of measure theory is this: How do we measure things, and what can we measure? What is measurable? Mathematically, this is harder to define than you would think.
Take a sphere, or ball, in Euclidean space. One possible measure that’s easy to explain is the Jordan measure. The Jordan measure basically decomposes everything into blocks. Think of it as the Minecraft measure. The reasoning behind it is that it’s easy to measure a block by just measuring its sides. Now, of course, a sphere cannot technically be decomposed into blocks. So what you need to do is to find the smallest blocks that will enclose the sphere from the outside, and the smallest blocks that will fill it up from the inside, in both cases without crossing the bounday. If the two measures coincide, then the sphere is Jordan-measurable.
As it happens, a sphere is Jordan-measurable. But many other things which should be measurable are not, which is why the Jordan measure was soon superseded by the Lebesque measure. But more on that some other day.