Wednesday, November 18, 2015

Another impossibility result for finitely additive probabilities and invariance

Consider a countably infinite sequence of fair and independent coin tosses. Given the Axiom of Choice, there is no finitely additive probability measure that satisfies these conditions:

  1. It is defined for all sets of outcomes.
  2. It agrees with the classical probabilities where these are defined.
  3. It is invariant under permutations of coins.
(Sketch of proof: Index the coins with members of the free group of rank two. The members of the group then induce permutations of coins, and hence act on the space of outcomes. The set of non-trivial fixed points under that action has classical probability zero. Throwing that out, we can use a standard paradoxical decomposition of the free group of rank two to generate a paradoxical decomposition of the rest of our space of results--here Choice will be used--and that rules out the possibility of a finitely additive probability measure.)


Alexander R Pruss said...

I think I can weaken the second condition to the claim that there is a non-zero probability that the tosses don't all come out the same.

Alexander R Pruss said...

And drop "fair and independent". :-)