There are definitely noncommutative analogues for Lusin's theorem and Egoroff's theorem (found in Blackadar for example). I'm curious if there is a version of the first principle:
Every Lebesgue measurable set is almost an open set.
In this noncommutative setting, I wonder if this is just saying that a von Neumann algebra is densely spanned by its projections (this is certainly true)... but is this the appropriate analogy?