spaces.

\[3.1\][@P] Let $\{M_n\}_{n\geq1}$ be a sequence of von Neumann subalgebras of $M$ such that $M_1\subset M_2\subset
\dots$. If one has $\overline{\bigcup\limits_{n}M_n}^{w^*}=M$ then the sequence $\{M_n\}_{n\geq1}$ is called a *filtration* of $M$.

Let $M$ be as in Section 4, and $\{M_n\}$ be a filtration of $M$. Let $\mathcal{E}(\cdot|M_n)$ be a conditional expectation from $L_p(M,\Phi)$ to $L_p(M_n,\Phi)$.

\[3.2\]A sequence $\{x_n\}\subset
L_p(M,\Phi)$ is called an *$L_p(M,\Phi)$-martingale* with respect to $\{M_n\}_{n\geq1}$ if one has $\mathcal{E}(x_{n+1}|M_n)=x_n$ for every $n\geq 1$.

\[3.3\] If $\{x_n\}\subset
L_1(M,\Phi)$ is an $L_1(M,\Phi)$-martingale with respect to $\{M_n\}_{n\geq1}$ then $\{x_n(\omega)\}\subset
L_1(M(\omega),\tau_\omega)$ is also martingale with respect to filtration $\{M_n(\omega)\}_{n\geq1}$ of $M_\infty(\omega):=\overline{\bigcup\limits_{n}M_n(\omega)}^{w^*}$ for almost all $\omega\in \Omega$.

Let $\mathcal{E}(x_{n+1}|M_n)=x_n$ for every $n\geq1$. According Theorem \[2.3\] we have $$\mathcal{E}_\omega(x_{n+1}(\omega)|M_n(\omega))=\mathcal{E}(x_{n+1}|M_n)(\omega)=x_n(\omega)$$ for almost all $\omega\in \Omega$, i.e. $\{x_n(\omega)\}\subset
L_1(M(\omega),\tau_\omega)$ is a martingale with respect to the filtration $(M_n(\omega))_{n\geq1}$ of $\overline{\bigcup\limits_{n}M_n(\omega)}^{w^*}$.

If for a *$L_p(M,\Phi)$-martingale* $\{x_n\}$ one has $\sup\limits_{n\geq 1}\|x_n\|_p \in L^0(\Omega)$ then $\{x_n\}$ is called a *bounded $L_p(M,\Phi)$-martingale*.

\[3.5\] Let $p>1$. The following statements hold:

1.  Let $x\in L_p(M,\Phi)$, $p\geq1$ and define $x_n=\mathcal{E}(x|M_n)$. Then $\{x_n\}$ is a bounded $L_p(M,\Phi)$-martingale, and $x_n\rightarrow x$ in $L_p(M,\Phi)$.

2.  Let $\{x_n\}$ be a bounded $L_p(M,\Phi)$-martingale. Then there exists $x\in L_p(M,\Phi)$ such that $x_n=\mathcal{E}(x|M_n)$.

(i). Due to $\|x_n\|_p=\|\mathcal{E}(x|M_n)\|_p\leq
\|x\|_p$ we 