The Besicovitch–Federer projection theorem is false in every infinite-dimensional Banach space
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We construct a purely unrectifiable set of finite H1-measure in every infinite-dimensional separable Banach space X whose image under every 0 ≠ x* ∈ X* has positive Lebesgue measure. This demonstrates completely the failure of the Besicovitch–Federer projection theorem in infinitedimensional Banach spaces.
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