Formalism of non-communitive (quantum) probability space 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? 
 A: What's a bit more recent is Section 3.2: Test Spaces in https://plato.stanford.edu/entries/qt-quantlog/ And you can google test spaces and/or generalized probability theory for much additional info. For example, https://arxiv.org/abs/quant-ph/0405178
That same Wilce article is a contribution in the book https://books.google.com/books?id=kdnNtbqRbiAC&pg=PA447&lpg=PA443 (you'll have to scroll back to page 443 for the title page; I can only seem to get it to display page 447). And that book contains a variety of other very relevant contributions that you can discern from its toc. While very expensive, many individual contributions can be legitimately downloaded from their authors' homepages (and I noticed an apparently pirated pdf of the whole book, but won't mention its url).
Re Gudder, yeah, that's a real old paper. He subsequently further developed his ideas in his own textbooks https://books.google.com/books?id=9qQaZHTQjxEC&pg=PA001
and https://books.google.com/books?id=mmjiBQAAQBAJ&pg=PA001 which are still pretty old (I'd guess the second, "Quantum Probability", is more likely the one you might want to take a closer look at). And somewhat (okay, lots:) off-topic, you simply must take a look at his http://sgudder.deviantart.com/gallery/
