Let $E$ be an $s$-rectifiable set in $\mathbb{R}^n$ of positive $s$-dimensional Hausdorff measure $H^s(E)>0$. The original question I have is:
Can there exist a one-to-one Lipschitz function $f\colon E\to \mathbb{R}^k$ where $k<s$? (In fact $f$ is the restriction of a linear mapping in my case)
If the set $E$ contains some (one-to-one) image of an open set of $\mathbb{R}^s$ we can easily see by Brouwers invariance of domain theorem that this is not possible. However rectifiable sets do not necessarily satisfy this condition even if they have positive Hausdorff measure.
Another approach I had was based on the coarea formula:
$$\int_{E}J_{f}^S(x) dH^{s}(x)= \int_{\mathbb{R}^k} H^{s-k}(E\cap f^{-1}(y))d\lambda^k(y)$$
Here, the right hand side is obviously zero because $f^{-1}(y)$ is at most a single point and the Hausdorff measure $H^{s-k}$ is not the counting measure because $s>k$. This implies that the Jacobian on the left hand side must be equal to zero $H^s$-almost everywhere on $E$. I have the feeling that this is not possible, but was not able to find any rigorous reason why this cannot happen for arbitrary rectifiable sets and Lipschitz functions.
Most results show that the Hausdorff dimension cannot be increased by Lipschitz functions, is there a similar result that it cannot be reduced by one-to-one Lipschitz functions? At least for rectifiable sets?
I have access to most standard literature on geometric measure theory (Ambrosio, Fusco, Pallara; Federer; Krantz&Parks; Mattila) but couldn't find any results on this problem. It seems to me like a trivial question but the missing of a concrete statement in the literature made me cautious to assume that this holds due to some trivial reason.