It has been argued that quantum probability requires a different treatment from other random events, and a different formalization is needed than that provided by Kolmogorov's axioms. And as a physicist by training I have read such statements dozens of times.
However, I have never seen a simple and concrete example showing that some result of quantum mechanics cannot be reduced to the usal treatment random events over a measure space $(\Omega, \mathcal{F},\mathbb{P})$.
Could someone provide a reasonable example of the probability used in quantum mechanics not being reducible to Kolmogorov's axioms?
Disclaimer: I am asking the question here and not in a physics space, because I would like to see a rigorous demonstration of the fact, with the formality and rigor proper to mathematics, so as not to leave room for the ambiguity typical of discussions in physics.