Probability and Quantum mechanics I don't quite understand how the probability language of sample spaces, $\sigma-$algebra, random variables, etc, fit into the quantum mechanics' formalism.  
To wit, we usually say that an observable is a linear operator in a Hilbert space, and afterwards we define the "expected value" of this operator. But what sense does this make in the probability framework? This is not a random variable in any sense. When we compute the expected value of position or moment, what is our sample space, our algebra, our probability measure?   
Also, it is always said that $\vert \psi \vert ^{2}$ gives the density probability function (with $\psi$ a solution of the Schrödinger equation)... the density of what random variable?  And what would be the density or distribution of the random variables "position" and "moment"?  Certanly, there must be such random variables. Or they only exist formally as operators in a Hilbert space? But, if so, What sense does it have, formally, to compute its expected value?
Thanks
 A: The probability model of quantum mechanics is different from the Kolmogorov model, is due to von Neumann, and actually predates the Kolmogorov model. The basic model goes something like this: events are represented by the lattice of projectors on a Hilbert space. The elementary outcomes are the one-dimenstional projectors. Probabilities are assigned to events $P$ as
$$\text{Prob}(P) = \omega(P)$$
with $\omega$ satisfying the following requirements


*

*$\omega(0)=0$

*$\omega(\mathbb{I})=1$ with $\mathbb{I}$ the identity.

*$\omega(\sum_j P_j) =\sum_j \omega(P_j)$ whenever $\{P_1,P_2,P_3,\ldots\}$ is an at most countable collection of mutually orthogonal projectors.


You can see that this is analogous to the Kolmogorov axioms. The functionals $\omega$ have a convex structure and the extreme states, called pure states, play the role of Dirac measures on the quantum probability space. 
Gleason's theorem then states that for a separable Hilbert space of dimension greater than or equal to three, any state on the lattice of projectors on that Hilbert space is a convex combination of pure states and any pure state is of the form
$$\text{Prob}(P)=\langle \phi , P \phi\rangle$$
with $\phi$ in the Hilbert space and $\|\phi\|=1$.
Now, the Spectral Theorem tells you how to compute the probabilities of observing an observable within a certain range of values $S$. If
$$A=\int_{\mathbb{R}}\lambda dE^A(\lambda)$$
in which $E^A$ is the spectral measure of $A$, then 
$$\text{Prob(Observed value of } A \text{ belongs to } S) = \langle \phi , E^A(S) \phi\rangle$$
Similarly, the expected value of $A$ can be written as
$$\langle A \rangle_{\phi} = \int_{\mathbb{R}}\lambda \langle \phi, dE^A(\lambda)\phi\rangle = \langle \phi, A \phi \rangle$$
which is the usual expectation formula we encounter in basic quantum mechanics textbooks.
Now, to see how the classical model fits in, take the triplet $(\Omega,\Sigma,\mu)$ to represent a usual Kolmogorov probability model. Then, $L^2(\Omega,\mu)$ is the space of complex-valued square integrable functions w.r.t. the measure $\mu$ forms a Hilbert space. The observables on this space are given by functions out of the same Hilbert space that act as multiplication operators on the Hilbert space:
$$\phi \mapsto (M_f \phi)(x):= f(x)\phi(x), \;\; \phi,f \in L^2(\Omega,\mu) \; .$$
The projectors are given by indicator functions on elements $S$ of the sigma algebra $\Sigma$:
$$I_S: x \mapsto \begin{cases}1 & \text{if } x \in S \\ 0 & \text{else}\end{cases}$$
or rather, a projector is an observable $M_{I_S}$.
The functional $\omega$ is defined as 
$$\omega(M_f) := \int_{\Omega} f(x) \mu(dx) = \langle {\bf 1}, f {\bf 1} \rangle  \; .$$
In which ${\bf 1}$ is the constant function, a normalized vector in $L^2(\Omega,\mu)$. Thus for a projector we have 
$$\omega(M_{I_S}) := \int_{\Omega} I_S(x) \mu(dx) = \mu(S) = \langle {\bf 1}, I_S {\bf 1} \rangle \; .$$
