Let , be Hausdorff measures on compact metric spaces X, Y and let and 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 -measure or -measure null, respectively. We shall show that if is not σ-finite, and one of the quotient Boolean algebras embeds densely in the other, then for some Borel B with , takes on Borel subsets of B only values 0 or ∞.
This is a particular instance of some more general results concerning Boolean algebras , where J is a σ-ideal of Borel sets generated by compact sets.