Multiplicative $A_\infty$-structure in terms of spectral sequences of fibrations

In the present paper, the technique of spectral sequences with $A_\infty$-structures in their terms is developed for differential algebras with filtrations. Applications of this technique to the multiplicative spectral sequences of fibrations are given. We show that if the base of fibration is connected and simply connected, then the structure graded $A_\infty$-algebra in the second term of the spectral sequence of a fibration is the tensor product of the cohomology $A_\infty$-algebra of the base and the cohomology $A_\infty$-algebra of the fibre of this fibration.