Abstract: We show that if X is a Banach space of infinite dimension and μ h is a Hausdorff measure, where h is continuous, then there exists a measurable set K ⊂ X such that 0<μ h ( K )< + ∞. We also characterize the normed spaces in which the unit ball can be covered by a sequence of balls whose radii r n < 1 converge to zero as n → ∞.