Is the Born rule indeed wrong? This is a question about the validity of a preprint, arXiv:quant-ph/0509089, which claims that the "Copenhagen Interpretation of QM is incorrect" (same title, authored by Guang-Liang Li and Victor O.K. Li, dated 23/09/2005).
This is a question about physics, and it can be answered. A reminder, whether the Born rule is correct or not is not a question which QM can answer. In other words, the fact that QM is correct, or consistent, or (in)complete, or that QM provides accurate predictions is irrelevant here. The Born rule (in my understanding) could be seen as an axiom to explain (in a certain way) the measurement problem in QM. It appears valid that such a hypothesis is inspected for consistency or validity, like the paper does. The paper claims (among other points) that the Born rule is not valid, and is based on false assumptions.
It would take me a long time to verify myself, step-by-step that a) the derivation of Li and Li is correct, and b) that it is not based on wrong assumption concerning Born's rule. 
So here is my question: has anybody studied the paper, and come to the conclusion that the paper's claims are either incorrect, or unfounded? 
 A: As StephenG mentioned in a comment, the paper you're asking about is the subject of a commentary in arXiv:quant-ph/0509130, by Markus Bier; Li and Li attempt a rebuttal of that comment in Appendix C of v2 of their paper.
The comment by Markus Bier is phrased in dry academic language, and there are certain aspects of the phrasing that simply do not fit within the format of an arXiv eprint. So, let me fill those back in:


*

*Theorem $1$ in Li and Li's paper is grossly incorrect, and it represents an almost-comical degree of lack of understanding of how mathematical probability works.


For those who cannot be bothered to test dig into the papers themselves, here's the offending theorem:

Theorem 1: For a probability space $(Ω, \mathcal F, P)$, there are values almost everywhere in $(0, 1)$ that the probability measure $P$ cannot take.

(Here, as usual $\mathcal F$ is a $\sigma$-algebra on $\Omega$, and $P:\mathcal F\to [0,1]$ is a probability measure.) As Bier points out, this is obviously wrong, and it is in explicit conflict with the basic examples constructed in any undergraduate-mathematics measure-theory class, with the usual Lebesgue-Borel measure on $\mathbb R$ being the most obvious example.
(Li and Li then go on to state, in their response, that the usual Lebesgue-Borel measure on $\mathbb R$ is "merely an assumption in the disguise of a definition" and that its construction "causes contradictions". Float that by the next measure theorist or mathematical probabilist you meet, and see what they think about it.)
The core error in the proof is in the third paragraph, and it boils down to the fact, pointed out by Markus Bier, that $G(\mathcal F)$ (as Li and Li define it) is not guaranteed to be a countable set. The implication is this: they attempt to argue that $\cup_{F\in G(\mathcal F)} \Phi(F)$ is contained in a union of sets with arbitrarily small measure, so it must have arbitrarily small measure. The claim would work if the union were countable, but it breaks down for uncountable unions like the one in use.
Frankly, by the time a paper has gotten this much basic stuff wrong, that's when you should stop reading. The paper is attempting to make a deep claim about the nature of physical reality, and it is attempting to bury the justification of that claim in enough esoteric-looking measure theory that physicists won't question it; and when someone did poke the obvious holes at their faulty arguments, they just attempted to pile more jargon on top of it.
So, to put it plainly: this paper offers no contribution of substance, and it can safely be ignored.
