How does one construct a probability space with orthogonal projectors on a Hilbert space? In this answer on physics stack exchange
https://physics.stackexchange.com/a/116609
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. 
 A: The thing you're looking for is a projection-valued measure.  See the last section of that link for the connection to quantum mechanics.
The most common way to construct such a measure is as the spectral measure of some self-adjoint operator (observable) on $H$.  This is what Valter is referring to when he says that this is just the spectral theorem in disguise.
I might be lazy and lie a bit, but I'll try to motivate the situation. 
We'll take the model of quantum mechanics where observables are self-adjoint operators on some Hilbert space $H$, where the unit ball of $H$ represents the set of states for our quantum system. Let's say you have an observable (like the position operator, call it $X$) whose measurements should spit out real values.  The spectral theorem then gives us a spectral measure from $X$, which is a projection-valued measure.  It takes in Borel subsets of the spectrum of $X$ (Borel subsets of $\mathbb{R}$) and spits out projections on $H$.  
If $E$ is a Borel subset of $X$'s spectrum, let's call the associated spectral projection $\pi(E) \in P(H)$.  The point then is that we should be able to talk about the probability that, for a given prepared state $\phi \in H$, $X$ will be measured to fall inside of $E$.  In this picture, the probability we want is given by $\langle \phi, \pi(E) \phi\rangle$.  So the spectral measure for a given observable gives us a way of computing the probability that our measurements will fall into a specific range (or, more generally, into a specific Borel subset of the spectrum).  
A: See this blog post. The term really should be "noncommutative probability space." 
