TITLE: A remark of Connes
QUESTION [33 upvotes]: In an interview (at http://www.alainconnes.org/docs/Inteng.pdf) Connes remarks that

I had been working on non-standard analysis, but after a while I had found a catch in the theory.... The point is that as soon as you have a non-standard number, you get a non-measurable set. And in Choquet’s circle, having well studied the Polish school, we knew that every set you can name is measurable; so it seemed utterly doomed to failure to try to use non-standard analysis to do physics.  

What does he mean; what is he referring to?

REPLY [5 votes]: A recent article by Leichtnam and myself (arxiv) in the American Mathematical Monthly contains a "theorem" to the effect that, in the presence of a construction of the hyperreals, the following is true: as soon as you have a Connes infinitesimal, you get a non-measurable set.