(a(\omega)x(\omega)b(\omega))&=&\lim\limits_{n\rightarrow\infty}\ell(a_n)(\omega)\ell(\mathcal{E}(x_n))(\omega)\ell(b_n)(\omega)\\[2mm]
&=&a(\omega)T_\omega (x(\omega))b(\omega).\end{aligned}$$

\(iii) Let $x(\omega)\in M(\omega)$. Then $$\tau_\omega(T_\omega
(x(\omega))=\tau_\omega(\lim\limits_{n\rightarrow\infty}\ell(\mathcal{E}(x_n)(\omega)=\lim\limits_{n\rightarrow\infty}p(\Phi(\mathcal{E}(x_n))(\omega)$$ for some $x_n\in M$. From $\Phi(\mathcal{E}(x_n))=\Phi(x_n)$ one gets $$\tau_\omega(T_\omega
(x(\omega))=\lim\limits_{n\rightarrow\infty}p(\Phi(x_n)(\omega)=\lim\limits_{n\rightarrow\infty}
\tau_\omega(\ell(x_n)(\omega))=\tau_\omega(x(\omega)).$$

Hence, $T_\omega$ is a conditional expectation from $M(\omega)$ onto $N(\omega)$. So, it can be extended to $T_\omega:
L_p(M(\omega),\tau_\omega)\rightarrow L_p(N(\omega),\tau_\omega)$, which is a conditional expectation from $L_p(M(\omega),\tau_\omega)$ onto $L_p(N(\omega),\tau_\omega)$, i.e. $T_\omega=\mathcal{E}_\omega.$

We show that $\mathcal{E}(x)(\omega)= \mathcal{E}_\omega(x(\omega))$ for all $x\in L_p(M,\Phi)$ and for almost all $\omega\in\Omega$. It is clear that $\mathcal{E}_\omega x(\omega) =
 \mathcal{E}(x)(\omega)$ for $x\in M$ and all $\omega\in\Omega$.

Let $x\in L_p(M,\Phi)$. As $M$ $(bo)$-dense in $L_p(M,\Phi)$, then there exists a sequence $\{x_n\}\subset M$ such that $\|x_n-x\|_p\stackrel{(o)}\rightarrow0.$ Then $\|x_n(\omega) -
 x(\omega)\|_{L_p(M(\omega),\tau_\omega)} \rightarrow0$ for almost all $\omega\in \Omega$. From the equality $\mathcal{E}(x)=(bo) -
\lim\limits_{n\rightarrow\infty}\mathcal{E}(x_n)$ we get $$\|\mathcal{E}_\omega(x_n(\omega)) -
 (\mathcal{E}x)(\omega)\|_{L_p(M(\omega),\tau_\omega)} \rightarrow0$$ for almost all $\omega\in \Omega$, i.e. $\lim\limits_{n\rightarrow\infty}\mathcal{E}_\omega(x_n(\omega))
 =
 (\mathcal{E}x)(\omega)$ for almost all $\omega\in \Omega$.

On the other hand, from the continuity of $\mathcal{E}_\omega$ we obtain $\lim\limits_n \mathcal{E}_\omega (x_n(\omega)) =
\mathcal{E}_\omega
 x(\omega)$ a.e. Hence $\mathcal{E}(x)(\omega)= \mathcal{E}_\omega(x(\omega))$ for all $x\in L_p(M,\Phi)$ and for almost all $\omega\in\Omega$. This completes the proof.

Martingales theorems in noncommutative $L_p(M,\Phi)$ spaces
===========================================================

In this section, as an application of the results of the previous section, we are going to prove the norm-convergence of martingales in the noncommutative $L_p(M,\Phi)$ 