In an old paper by Gudder http://www.ams.org/journals/proc/1969-021-02/S0002-9939-1969-0243793-1/S0002-9939-1969-0243793-1.pdf,
he defined: "a quantum probability space is a triple $(\Omega, C, M)$ where $C$ is a $\sigma$-class of subsets of $\Omega$ and $M$ is the set of states on $C$."
What does a $\sigma$-class means in math? I understand what a $\sigma$-field or algebra is.
What is "states" on $C$? Is $m\in M$ a mapping that $m:C\to[0,1]$?
I wonder if there are better (newer) notes or books on the formalism of quantum probability.
A paper https://arxiv.org/pdf/quant-ph/0601158.pdf explained the quantum probability space in detail. Are those concepts well-accepted?