Extending Borel’s conjecture from measure to dimension

We study subsets A A of R N {R}^N , gauge functions f f and their associated Hausdorff measures H f H^f . A set A ⊆ R N A⊆ {R}^N has strong dimension f f if and only if H f > 0 H^f>0 and for all gauge functions h h of higher order than f f , H h ( A ) = 0 H^h(A)=0 . We show that it is consistent with the axioms of set theory that if A A has strong dimension f f then A A is σ σ -finite for H f H^f .