Commutativity of the Born Rule

[the_pulp]

is the probability that a system in state y before the measurement will be in state x after the measurement, at least if we're talking about a measurement that doesn't destroy the system.

Yes, that's how I see it.

This is correct. However, vanhees71 wasn't talking about the dual of the Hilbert space. He was talking about the dual of one of its vector subspaces. If K⊂H, then H∗⊂K∗. The dual of the subspace is larger than the dual of the Hilbert space. It's a topological vector space, but not a Hilbert space. Members of that space may not have a norm for example. This trick allows us to assign a meaning to "eigenstates" of operators like position and momentum. They can be defined as members of K∗−H∗.

(Not sure if this has anything to do with the topic of this thread. I'm just making an observation about the mathematics).

Yes I've just reading Ballentine again, but as you said, its true but not related to the original question, to me.