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In the limit theory for strictly stationary processes $f\circ T^i,i\in \mathbb{Z}$, the decomposition $f=m+g-g\circ T$ proved to be very useful; here $T$ is a bimeasurable and measure preserving transformation an $(m\circ T^i)$ is a martingale difference sequence. We shall study the uniqueness of the decomposition when the filtration of $(m\circ T^i)$ is fixed. The case when the filtration varies is solved in [13]. The necessary and sufficient condition of the existence of the decomposition were given in [12] (for earlier and weaker versions of the results see [7])....