fa.functional analysis – Quick derivation of classical probability theory from von Neumann algebraic framework

Watching (the begining of) a lecture on free probability theory by Dimitri Shlyakhtenko https://www.youtube.com/watch?v=F8Urtr39jM0, I’m led to the following question

Question. How can one build classical probability theory (measurable functions on measure space, expectations, etc.) from the framework of “von Neumann algebras” ?

Note. I’d really appreciate a bottom-up answer (not just a dry stack of formal encyclopedic statements, except perhaps when strictly needed). Thanks in advance!