Let $M$ be a semifinite von Neumann algebra with a faithful normal semifinite trace $\tau$. For any projection $P\in M$ with $\tau(P)=\infty$, we can always find an increasing net $P_\alpha$ with $\tau(P_\alpha)<\infty$ such that $P_\alpha \uparrow P$. Since $\tau(P)=\sup_\alpha\tau(P_\alpha)$, we can find a set $\{P_n\}$ of projections from $\{P_\alpha\}$ with $\tau(P_n) \ge n$.
Then, for every $P_\alpha$, we always can find an $n$ such that $\tau(P_n)\ge n > \tau(P_\alpha)$ (i.e. $P_n \ge P_\alpha$). So, $\{P_n\}$ is a subnet of $\{P_\alpha\}$. Therefore $P_n\uparrow_n P$.
Well, so, can I say that for every projection in a semifinite von Neumann algebra, we can always find an increasing sequence of $\tau$-finite projections tending to this projection? I have never seen something like that. I think probably I am wrong. Can anyone explain a little bit for me?
Now, I have realized that the problem is the net is not totally ordered!