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The classical Besicovitch-Federer projection theorem implies that the d-dimensional Hausdorff measure of a set in Euclidean space with non-negligible d-unrectifiable part will strictly decrease under orthogonal projection onto almost every d-dimensional linear subspace.
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J. Harrison and H. Pugh, Solutions to the reifenberg plateau problem with cohomological spanning conditions , Calculus of Variations and Partial Differential Equations 55
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