Fix $ k \in \mathbb{N} $ and let $ H^k $ be the $k$-dimensional Hausdorff measure on $\ell^\infty $. Also, if $ V $ is a subspace of $ \ell^\infty $, we denote the projection onto $ V $ by $ \pi_V $. Let $ A $ be a subset of $ \ell^\infty $ such that $ H^k(\pi_V(A)) = 0 $ for all **finite** dimensional subspaces $ V $ of $ \ell^\infty $.

Can we conclude that $ H^k(A) = 0 $?

Any help would be appreciated.

Thanks.

for any projector $\pi$ with finite dimensional rangeallows a positive answer? $\endgroup$