Yesterday, I discovered an error in the proof of “Theorem 1” of this recent paper of mine (arxiv version). The error occurs in the harder direction of Lemma 2. I do not know how to fix the error. Here’s what I know to remain of the “Theorem”. The proof that (i) implies (ii)–(v) is unaffected. The proof that (iv) implies (ii)–(v) is also unaffected, and likewise unaffected is the equivalence of (ii), (iii) and (v).
But I no longer know if any of (ii)–(v) imply (i). However, (i) is true under the stronger assumption that G is supramenable or that there exist invariant hyperreal probabilities.
The above remarks suffice for almost all the philosophical points in the paper (the philosophical point that behavior for countable sets is decisive is no longer supported in the full conditional probability case), and all the applications I mention in the paper.
I do not know if “Theorem 1” is true. This is an interesting mathematical question.
Update: The error has been fixed and Theorem 1's proof now works.
2 comments:
I am making some progress on fixing the error. I am not 100% sure my fix works, but it seems to.
In happy news, my fix DOES work: I've proved Lemma 2, and the proof was verified by a mathematician who works in the field. In fact, my new proof shows the interesting fact that if any of the conditions of Theorem 1 holds, then any unconditional G-invariant probability on Omega can be extended to a full conditional one.
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