How does one construct a probability space with orthogonal projectors on a Hilbert space? In this answer on physics stack exchange < Valter Moretti provides a very in-depth understanding as to why Boolean algebras are insufficient for quantum mechanics. If I understand him correctly, he then is saying that the usual $\sigma$-algebra isn't appropriate for quantum mechanics and, consequently, a new mathematical toolkit is needed. Based on the post, I conclude that this type of probability space is a triple $(\Omega, \mathcal{P}(H),\Bbb P)$ where $\mathcal{P}(H)$, the orthogonal projectors of a separable Hilbert space $H$, replace the usual $\mathcal{F}$ $\sigma$-algebra. Can someone provide a formal definition for this kind of probability space as well as the term for it? I attempted to find a formal mathematical definition but wasn't successful. If am I being naive and $\mathcal{P}(H)$ is a type of $\sigma$-algebra, please explain why it is one.

See this blog post. The term really should be "noncommutative probability space."