the set of all measurable operators. If $M$ is a commutative von Neumann algebra, it is $*$-isomorphic to the $*$-algebra $L_\infty(\Omega,\Sigma,\mu)$ of all essentially bounded complex measurable functions with the identification almost everywhere, where $(\Omega,\Sigma,\mu)$ is a measurable space. In addition $S(M)\cong L_0(\Omega,\Sigma,\mu)$ [@S].

Let $x,y$ be measurable operators. Then $x+y,~xy$ and $x^*$ are densely-defined and preclosed. Moreover, the closures $\overline{x+y}$ (strong sum), $\overline{xy}$ (strong product) and $x^*$ are also measurable, and $S(M)$ is a $*$-algebra with respect to the strong sum, strong product, and the adjoint operation (see [@S]). For any subset $E\subset S(M)$ we denote by $E_{sa}$ (resp. $ E_+$ ) the set of all self-adjoint (resp. positive ) operators from $E.$

For $x\in S(M)$ let $x=u|x|$ be the polar decomposition, where $|x|=(x^*x)^{\frac{1}{2}},$ $u$ is a partial isometry in $B(H).$ Then $u\in M$ and $|x|\in S(M).$ If $x\in S_h(M)$ and $\{E_\lambda(x)\}$ are the spectral projections of $x,$ then $\{E_\lambda(x)\}\subset P(M).$

The locally measure topology $t(M)$ on $L_0(\Omega,\Sigma,\mu)$ is by definition the linear (Hausdorff) topology whose fundamental system of neighborhoods of $0$ is given by $$W(B,\varepsilon,\delta)=\{f\in\ L_0(\Omega,\, \Sigma,\, \mu)
\colon \hbox{ there exists a set } \ E\in \Sigma, \mbox{ such
that}$$ $$\ E\subseteq B, \ \mu(B\setminus E)\leq\delta, \
