On Boolean algebras related to σ-ideals generated by compact sets

Let μhμh, μgμg be Hausdorff measures on compact metric spaces X, Y   and let Bor(X)/Jσ(μh)Bor(X)/Jσ(μh) and Bor(Y)/J0(μg)Bor(Y)/J0(μg) be the Boolean algebras of Borel sets modulo σ-ideals of Borel sets that can be covered by countably many compact sets of σ  -finite μhμh-measure or μgμg-measure null, respectively. We shall show that if μhμh is not σ-finite, and one of the quotient Boolean algebras embeds densely in the other, then for some Borel B   with μh(B)=∞μh(B)=∞, μhμh takes on Borel subsets of B only values 0 or ∞.This is a particular instance of some more general results concerning Boolean algebras Bor(X)/JBor(X)/J, where J is a σ-ideal of Borel sets generated by compact sets.