If $M\subset B(H)$ is a semifinite von Neumann algebra with faithful, normal, semifinite trace $\tau$, then a closed operator $T:H\rightarrow H$ intertwining the action of $M$ is $\tau$-premeasurable if for every $\delta>0$ there is a projection $p\in M$ such that $pH\subset dom(T)$ and $\tau(1-p)<\delta$.

This roughly means that it is possible to find $M$-submodules inside $dom(T)$ whose module projection complements are "$\tau$-small", and so bad behavior can be contained inside the small piece in order to imitate integration theory techniques.

Of course this relies really strongly on the availability of the semifinite trace. Does this notion of measurability have a natural generalization/analogue that holds for all von Neumann algebras? If it does, then where is the best place to look for it in the literature?

(I am not optimistic, since I expect there to be difficulties like those one finds when passing from the semifinite to general case with Noncommutative $L^p$-spaces. I hope that I am wrong, though, and some nice gem is hidden somewhere that will give access to this particular notion/theme in the general setting.)