The above lemma is from Takesaki's book (Vol 2, Chapter VIII). I have two questions.
1.How to conclude that $e_nm_{\varphi}e_n$ is $\sigma$-weakly dense in $e_nMe_n$?
From Lemma 2.4, we only know that $m_{\varphi}$ is a $M_a^{\varphi}$-bimodule.
2.How to prove that $\cup e_nMe_n$ is $\sigma$-weakly dense in $M$?
The assumption is that $\varphi$ is fns. Being semifinite, there exists projections $\{p_j\}$ with $\varphi(p_j)<\infty$ for all $j$ and $p_j\to 1$ weakly. Then $p_j\in m_\varphi$ for all $j$. As $m_\varphi$ is an $M_a^\varphi$-module, given $y\in M_a^\varphi$ we have $a=\lim p_ja\in \overline{m_\varphi}$. As $M_a^\varphi$ is dense in $M$ (Lemma 2.3), we get that $$ \overline{m_\varphi}=\overline{M_a^\varphi}=M. $$