Proof. (1) $\Rightarrow$ (2): Let $H$ be a separable Hilbert space with $M \subset B(H)$ and $D \subset H$ be countable and dense. By the Arzelà-Ascoli Theorem, the strong operator topology on $\ol{B_M(0, 1)}$ coincides with the topology of pointwise convergence on $D$. By Theorem 6.3.10 and Proposition 5.5.7, $H^{D}$ is metrisable and separable. Therefore $\ol{B_M(0, 1)}\subset H^{D}$ is also metrisable and separable.

(2) $\Rightarrow$ (3): By Proposition 45.1.4.

(3) $\Rightarrow$ (4): By Lemma 43.4.10, the standard representation $(L^{2}(M, \tau), \pi, 1_{M})$ is faithful and normal. As $M$ is dense in $L^{2}(M, \tau)$, $\ol{B_M(0, 1)}$ being separable in $L^{2}(M, \tau)$ implies that $L^{2}(M, \tau)$ is separable.$\square$