I take minor issue with the phrase "correct" here. Thats one way you can do things but its also works completely fine to not do that. Another way of setting the

I take minor issue with the phrase "correct" here. Thats one way you can do things but its also works completely fine to not do that. Another way of setting these things up has your states be honest elements of L2, and says observables are just POVMs (i.e. maps from a space of measurable sets to positive operators which obey some natural restrictions like additivity). Then given a measurable subset A of the spectrum of some operator the Born-rule probability is just given by an inner product like < phi | P phi> where P is the projector you get if you integrate the spectral measure of the operator over A. This has the advantage of not having any funky "rigged" states suddenly appearing in your calculations and is also exactly how we deal with non-projective measurements in finite dimensional quantum mechanics. See here, for example https://en.wikipedia.org/wiki/POVM https://en.wikipedia.org/wiki/POVM