Abstract
We construct a purely unrectifiable set of finite H 1-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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Bate, D., Csörnyei, M. & Wilson, B. The Besicovitch–Federer projection theorem is false in every infinite-dimensional Banach space. Isr. J. Math. 220, 175–188 (2017). https://doi.org/10.1007/s11856-017-1514-y
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DOI: https://doi.org/10.1007/s11856-017-1514-y