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



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> **Question.** How can one build classical probability theory (measurable functions on measure space, expectations, etc.) from the framework of "von Neumann algebras"?
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**Note.** I'd really appreciate a bottom-up answer (not just a dry stack of formal encyclopedic statements, except perhaps when strictly needed).



Quick derivation of classical probability theory from von Neumann algebraic framework
At the beginning of his talk [What actually is free probability theory?](https://www.youtube.com/watch?v=IewAq-RNJFI) Tobias Mai explains how classical probability theory fits into the context of non-commutative probability theory.


