For a mapping $f: \Omega\to \bf{R}^n$, what kind of condition ensures that the onedimensional Hausdorff measure of $f^{1}(E)$ is zero whenever $E$ is of zero onedimensional Hausdorff measure zero. Note that f is not assumed to be a homeomorphism.

There may be a name for this, but it seems like a strange condition. Such a function cannot take a constant value on any set of positive Lebesgue measure, otherwise the inverse image of that constant (having zero 1D Hausdorff measure in the range) would have positive Lebesgue measure, and therefore infinite 1D Hausdorff measure. A good start might be to investigate the situation on maps $f:[0,1] \to \mathbb{R}$ with the Lebesgue measure in both places. There is also a related notion, called Lusin's N property, which means $f$ takes sets of measure zero into sets of measure zero (as opposed to $f^{1}$, as you desire). This is a quality of Lipschitz functions that Sobolev functions also inherit, and is necessary to satisfy the fundamental theorem of Calculus (along with being differentiable a.e., etc.). 

