We give an example, in the infinite dimensional separable Hilbert space, of a purely unrectifiable Borel set with finite nonzero one dimensional Hausdorff measure, whose projection is nonnegligible in a set of directions which is not Aronszajn null.
"An example pertaining to the failure of the Besicovitch-Federer structure Theorem in Hilbert space." Publ. Mat. 61 (1) 153 - 173, 2017. https://doi.org/10.5565/PUBLMAT_61117_06