y(\omega)$ in $L_p(M(\omega),\tau_\omega)$ for almost all $\omega\in \Omega$. Hence $\|x_n-y\|_p\stackrel{(o)}\rightarrow0.$ This completes the proof.

Acknowledgments {#acknowledgments .unnumbered}
===============

The first author (I.G) acknowledges the MOHE Grant FRGS13-071-0312.

[99]{}

Albeverio S, Ayupov Sh A., Kudaybergenov K K, Non commutative Arens algebras and their derivations. J. Funct. Anal. 2007; 253: 287–302.

C. Cecchini, D. Petz, Norm convergence of generalized martingales in $L^p$-spaces over von Neumann algebras, *Acta Sci. Math. (Szeged)* **48** (1985), 55-63.

V.I. Chilin. I.G. Ganiev, An individual ergodic theorem for contractions in the Banach–Kantorovich lattice $L^p(\nabla,\mu)$. *Russian Math.* [**44**]{}(2000), 77–79.

V.I. Chilin, A.A. Katz, On abstract characterization of non-commutative $L_p$-spaces associated with center- valued trace, *Methods Funct. Anal. Topol.* [**11**]{}(2005), 346-–355

V.I. Chilin, B.S. Zakirov, Non-commutative $L_p$-spaces associated with a Maharam trace,*J. Operator Theory* [**68**]{} (2012), 67–83.

V.I. Chilin, B.S. Zakirov, Maharam traces on von Neumann algebras, *Methods Funct. Anal. Topol.* [**16**]{}(2)(2010), 101–111.

N. Dang-Ngoc, Pointwise convergence of martingales in von Neumann algebras, *Israel J. Math.* [**34**]{} (1979), 273–280.

J. Dixmier, *Les $C^*$- -algebres et leurs representations*, Paris, Gauthier - Villars Editeur, 1969.

I.G. Ganiev, The martingales convergence in the Banach-Kantorovich’s lattices $L_p(\widehat{\nabla},\widehat{\mu}),$ *Uzb. Math. Jour.* 2000, N.1, 18–26.

I.G. Ganiev, V.I. Chilin, Measurable bundles of noncommutative $L^p$-spaces associated with 