Noncommutative version of Littlewood's First Principle 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, we will most certainly have to use the fact that a von Neumann algebra is densely spanned by its projections ... but what precisely is Littlewood's First Principle.
 A: Here is one half of an answer, but it was too long for a comment. I'll edit if I manage to finish it...
When you have a $C^*$ algebra $C$ included in a von Neumann algebra $B$ ( and you might want to assume that $C$ is weakly dense in $B$) you have a notion of ``open projection" which are essentially the projections of $B$ which are spectral projection of element of $C$ corresponding to open interval of $\mathbb{R}$. (If the $C^*$ algebra is not separable, you might want to replace this definition by "the projections which are supremums of positive elements of $C$")
If the algebra $B$ is well chosen (for example, if $B$ is the enveloping von Neumann algebra of $C$, or its atomic part) then open projections are in bijection with closed left ideal of $C$. In general, this might not be the case, for exemple if $C=B$ every projections is open and they are in bijection with the weakly closed left ideals. In commutative terme, this is because ideals correspond to open subsets while open projection corresponds to open subset modulo equality almost everywhere.
These notions are discussed in detail in Akemann The general Stone-Weierstrass problem
You can formulate the principle as: "for any normal state on $B$, any projection $P \in B$ and any $\epsilon >0$ there is an open projection $P'$ such that $\eta(|P-P'|)<\epsilon$." Or maybe, "every projection can be approximated in the weak topology by open projections."
I'm not claiming that this will holds in general. I have to think more about it first. But maybe someone else will do it first.
