The Besicovitch-Federer projection theorem is false in every infinite dimensional Banach space (1506.08431v1)

Published 28 Jun 2015 in math.FA

Abstract: We construct a purely unrectifiable set of finite $\mathcal H^1$-measure in every infinite dimensional separable Banach space $X$ whose image under every $0\neq x^*\in X^*$ has positive Lebesgue measure. This demonstrates completely the failure of the Besicovitch-Federer projection theorem in infinite dimensional Banach spaces.

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