Concrete representation of atomic $(F_4)$ filtrations

We prove that for any sequence of functions adapted to a biparameter atomic filtration satisfying $(F_4)$ condition there is a sequence having the same joint distribution but adapted to the canonical $(F_4)$ filtration. Even in one parameter case our result is an improvement of the theorem due to Montgomery-Smith, since the construction gives a morphism of filtrations and does not depend on underlying sequence.