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Last year on this blog, I sketched out a non-rigorous probabilistic argument justifying the following well-known theorem:
Theorem 1. (Non-measurable sets exist) There exists a subset of the unit interval which is not Lebesgue-measurable.
The idea was to let E be a “random” subset of . If one (non-rigorously) applies the law of large numbers, one expects E to have “density” 1/2 with respect to every subinterval of , which would contradict the Lebesgue differentiation theorem.
I was recently asked whether I could in fact make the above argument rigorous. This turned out to be more difficult than I had anticipated, due to some technicalities in trying to make the concept of a random subset of (which requires an uncountable number of “coin flips” to generate) both rigorous and useful. However, there is a simpler variant of the above argument which can be made rigorous. Instead of letting E be a “random” subset of , one takes E to be an “alternating” set that contains “every other” real number in ; this again should have density 1/2 in every subinterval and thus again contradict the Lebesgue differentiation theorem.
Of course, in the standard model of the real numbers, it makes no sense to talk about “every other” or “every second” real number, as the real numbers are not discrete. If however one employs the language of non-standard analysis, then it is possible to make the above argument rigorous, and this is the purpose of my post today. I will assume some basic familiarity with non-standard analysis, for instance as discussed in this earlier post of mine.