Densities and liftings for derived algebras

Motivated by the construction of the algebra of Jordan measurable sets from the algebra of measurable sets in Euclidean spaces, we determine a natural class of structures ( X , S , A , I ) , where ( X , S ) is a topological space, A is an algebra of subsets of X, and I is an ideal of A , so that the derived structure ( X , S , ∂ A , ∂ I ) given by ∂ A : = { E ∈ A : ∂ E ∈ I } and ∂ I : = ∂ A ∩ I belongs to the same class. We provide a characterization of the derived structures whose ideal contains no nonempty open sets and derive from it that each such structure has a natural strong density and (assuming completeness) a strong lifting.