I am reading a book "C*-algebra and finite-Dimensional Approximations". In the fundamental facts, it introduce the Noncommunicative Lusin's theorem:
Let $A\in B(H)$ be a nondegenerate C*-algebra with $A''=M$. For every finite set of vectors $F\in H, \varepsilon>0$, projection $p_{0}\in M$ and self-adjoint $y\in M$, there exists a self-adjoint $x\in A$ and a projection $p\in M$ such that $p\leq p_{0}$, $||p(h)-p_{0}(h)||<\varepsilon$ for all $h\in F$, $||x||\leq min\{2||yp_{0}||,||y||\}+\varepsilon$ and $xp=px$.
Does the theorem above have some relations to the Lusin's theorem in real analysis? Could someone explain it for me? Thanks.
You have a typo at the end of your question: it should say $xp=yp$. This is similar to Lusin's theorem in the sense that in the original theorem you have the continuous functions hanging aroudn betweent the measurable functions, and off a set of small measure, you can get your measurable function equal to the continuous function.
Here you have to think of $A$ as the "continuous functions", and $y$ is your measurable function. If $\varepsilon$ is small enough, the inequality will force $p=p_0$ on the subspace spanned by $F$. In other words, given any finite-dimensional subspace, you can find $x\in A$ with $x=y$ on the subspace.
So, this theorem has the same flavour as Lusin's, in that your "bad" operator is equal to a "nice" one over some set.