>>124 つづき Recently though I stumbled upon the page and I see Terrence Tao's comment, where I copied the relevant paragraph,
If we have an infinite number of prisoners, with the hats assigned randomly (thus, we are working on the Bernoulli space ZN2), and one uses the strategy coming from the axiom of choice, then the event Ej that the jth prisoner does not go free is not measurable, but formally has probability 1/2 in the sense that Ej and its translate Ej+ej partition ZN2 where ej is the jth basis element, or in more prosaic language, if the jth prisoner’s hat gets switched, this flips whether the prisoner gets to go free or not. The “paradox” is the fact that while the Ej all seem to have probability 1/2, each element of the event space lies in only finitely many of the Ej. This can be seen to violate Fubini’s theorem ? if the Ej are all measurable. Of course, the Ej are not measurable, and so one’s intuition on probability should not be trusted here.
It feels like he concludes the non-measurability of Ej from a violation of Fubini, but I don't see it. Can someone flesh this argument out for me? It has been nagging me for a long time now and I would be very grateful :)
