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.