Elsevier

Advances in Mathematics

Volume 297, 16 July 2016, Pages 196-213
Advances in Mathematics

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

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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.

MSC

03E15
54H05
28A78

Keywords

Borel mapping
σ-ideal
Hausdorff measure

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