The Besicovitch-Federer projection theorem is false in every infinite dimensional Banach space

TL;DR

This paper constructs a special unrectifiable set in every infinite dimensional Banach space that defies the Besicovitch-Federer projection theorem, showing the theorem's failure in such spaces.

Contribution

It provides a counterexample demonstrating the failure of the Besicovitch-Federer projection theorem in all infinite dimensional Banach spaces.

Findings

Constructs a purely unrectifiable set with finite measure in every infinite dimensional Banach space.

Shows the image of this set under any non-zero dual element has positive Lebesgue measure.

Demonstrates the universal failure of the projection theorem in infinite dimensional settings.

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.