Monday, 14 November 2011

A simpler example of a nonmeasurable set

I don't claim that I'm the first person to have come up with this example; I only claim that yesterday is the first time I thought of it. This is a rather sorry state of affairs, because it's an obvious and nice example.

First, some background, the example of a nonmeasurable set that I was given when I first learnt measure theory went basically as follows. Start with $[0,1]$, and call $x,y\in[0,1]$ equivalent if $x-y\in{\bf Q}$. This defines an equivalence relation, and thus we can define a set $Z$ to consist of exactly one representative from each equivalence class. Then the interval $[0,1]$ is just the disjoint union of translates mod $1$ of $Z$ by rational numbers mod $1$ (or something). Because the rationals in $[0,1]$ are countably infinite this is incompatible with $Z$ being measurable: if $\mu(Z)>0$ then then $\mu([0,1])=\infty$, and if $\mu(Z)=0$ then $\mu([0,1])=0$.

This example is fine and beautiful and all that, but I find it rather hard to remember and especially difficult to visualize. Thus, yesterday, when I was explaining to my girlfriend in an intuitive way why not all sets are measurable, the example I produced was, by accident, not the above example.

From an intuitive point of view, if we're talking about $[0,1]$ and we're generally thinking about mod $1$ then we're probably better off just talking about the circle $S^1$ and not confusing the issue. Moreover, now translation mod $1$ is just rotation, which is much easier to visualize and, importantly, much easier to believe is measure-preserving. Now we just need to decompose $S^1$ into a countably infinite collection of copies of some set $Z$. Towards this end, let $T$ be any irrational rotation of $S^1$. (When I was talking yesterday, I started with the concrete example of rotation by $1$ degree, and then hastily changed this to rotation by $1$ radian.) Then for any point $x$ we can consider the orbit $\{T^n(x)\}_{n\in\mathbb{Z}}$, and we thus decompose $S^1$ into a union of orbits under the action of $T$. Let $Z$ be a set consisting of exactly one representative of each of these orbits. Since every orbit is infinite, $S^1$ is the disjoint union of
\[\ldots,T^{-2}(Z), T^{-1}(Z), Z, T(Z), T^2(Z),\ldots.\]
Since $T$ is measure-preserving, we have a contradiction as before.

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