Abstract
Let μh, μg be Hausdorff measures on compact metric spaces X, Y and let Bor(X)/Jσ(μh) and 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 μh-measure or μg-measure null, respectively. We shall show that if μh 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 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)/J, where J is a σ-ideal of Borel sets generated by compact sets.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
