Converse to the Alberti-representation criterion via \u007eD(n) sets

Establish whether the converse of Corollary 3.5 holds: if a measure μ on a compact metric space X has n+1 independent Alberti representations, then every set in \u007eD(n) (i.e., a set that is null with respect to curve fragments quantitatively transversal to certain Lipschitz maps into R^k with k ≤ n) must be μ-null.

Background

Corollary 3.5 proves one direction: if every \u007eD(n) set is μ-null, then μ has n+1 independent Alberti representations. This uses an extension procedure that constructs additional independent Alberti representations, building on Proposition 3.4.

The authors note that the converse implication is not established, leaving open whether possessing n+1 independent Alberti representations forces all \u007eD(n) sets to be μ-null.