Can POVM measurements be explained by projective measurements?

[A. Neumaier]

I make fewer explicit assumptions, but the assumptions are implicitly there. That's because I think like physicist, not mathematician, so I try to explain how nature works, not to present a mathematical proof. I don't make explicit assumptions which seem obvious to me from a physical point of view, because I see such details as distraction from really important ideas. But I perfectly understand that you, as a mathematician, don't like this type of reasoning.

I am not requiring mathematical rigor. But physics has a long and successful tradition in giving proofs at a high level, a level missing in your paper. You make suggestions, which might motivate why the results could possibly be true, but fail to give arguments that would satisfy anyone who really wants to understand what is behind. Your references to the literature only cover conditions where much stronger assumptions are needed to actually make a convincing justification.

In any case, nothing in your paper except for a few passing words relate to POVMs, thus making a further discussion of it off-topic in this thread.