The Stone representation of an atomic complete Boolean algebra is Marczewski-Burstin representable

In 1935, E. Marczewski defined the families s = {A ⊂ X : (∀P ∈ F)(∃Q ∈ F)(Q ⊂ A ∩ P or Q ⊂ P / A)} and s0 = {A ⊂ X : (∀P ∈ F)(∃Q ∈ F) Q ⊂ P / A} where F is the family of perfect sets and X is a Polish space. We say that the pair <Α, Λ> (where Α is the algebra of subsets of X ≠ ∅ and Α ⊃ Λ the ideal of sets) has MB-representation if there exists a family ∅ ≠ F ⊂ P(X) \ {∅} such that Α = S(F) and S0(F), where and are constructed analogously to s and s0. We will use two theorems published in [J. Appl. Anal. 9 (2003), 275-286] and [Bull. Pol. Acad. Sci. Math. 53 (2005), 239-250], to prove the theorem which is stated in the title.