get $$\sup\limits_{n\geq 1}\|x_n\|_p\leq \|x\|_p\in
L^0(\Omega).$$ According Theorem \[2.3\] one has $$x_n(\omega)=\mathcal{E}(x|M_n)(\omega)=\mathcal{E}_\omega(x(\omega)|M_n(\omega))$$ for almost all $\omega\in \Omega$. Then using Proposition 2.10 (i) [@P] we obtain $$\|x_n(\omega)-x(\omega)\|_{L_p(M(\omega),\tau_\omega)}\rightarrow
 0$$ for almost all $\omega\in \Omega$. From $$\|x_n-x\|_p(\omega)=\|x_n(\omega)-x(\omega)\|_{L_p(M(\omega),\tau_\omega)}$$ it follows that $\|x_n-x\|_p\stackrel{(o)}\rightarrow0$ i.e. $x_n\rightarrow x$ in $L_p(M,\Phi)$.

(ii). Let $x_n\in L_p(M,\Phi)$ be an $L_p(M,\Phi)$-martingale, i.e. $a:=\sup\limits_{n\geq 1}\|x_n\|_p \in
L^0(\Omega)$. Then $$\sup\limits_{n\geq
1}\|x_n(\omega)\|_{L_p(M(\omega),\tau_\omega)} =\sup\limits_{n\geq
1}\|x_n\|_p(\omega)=a(\omega)<\infty$$ for almost all $\omega\in
\Omega$. According Proposition 2.10 (ii) [@P] there exists $x(\omega)\in L_p(M(\omega),\tau_\omega)$, such that $x_n(\omega)=\mathcal{E}_\omega(x(\omega)|M_n(\omega))$ for almost all $\omega\in \Omega$. Again by Proposition 2.10 (i) [@P] we have $\|x_n(\omega)-x(\omega)\|_{L_p(M(\omega),\tau_\omega)}\rightarrow0$ for almost all $\omega\in \Omega$. This mean $x\in M(\Omega, X)$. Therefore, $x\in L_p(M,\Phi)$. So $$x_n=\widehat{\mathcal{E}_\omega(x(\omega)|M_n(\omega))}=\mathcal{E}(x|M_n).$$ This completes the proof.

\[3.6\] Let $p>1$. The following statements hold:

1.  Let $x_n\rightarrow x$ in $L_p(M,\Phi)$, then $\mathcal{E}(x_n|N)\rightarrow
    \mathcal{E}(x|N)$ in $L_p(M,\Phi)$ for any von Neumann subalgebra $N$ of $M$.

2.  If $x_n\rightarrow x$ in $L_p(M,\Phi)$ and $\overline{\bigcup\limits_{m}M_m}^{w^*}=M$, then $\mathcal{E}(x_n|M_m)\rightarrow x$ in $L_p(M,\Phi)$ as $n,m\rightarrow\infty$.

\(i) follows from Theorem \[2.3\].

(ii). It is clear that $$\begin{aligned}
\label{36-1}\|\mathcal{E}(x_n|M_m)-x\|_p&=&\|\mathcal{E}(x_n|M_m)-\mathcal{E}(x|M_m)+\mathcal{E}(x|M_m)-x\|_p\nonumber\\[2mm]
&\leq& \|x_n-x\|_p+\|\mathcal{E}(x|M_m)- x\|_p.\end{aligned}$$ By condition we have $\|x_n-x\|_p\stackrel{(o)}\rightarrow0$, and Theorem \[3.5\] (i) implies $\|\mathcal{E}(x|M_m)-
x\|_p\stackrel{(o)}\rightarrow0$. Hence, from one finds $\mathcal{E}(x_n|M_m)\rightarrow x$ in $L_p(M,\Phi)$ as $n,m\rightarrow\infty$.

Now we are going to prove the convergence of weighted averages of martingales 