Let $X$ be a $k+1$ rectifiable set with finite $k+1$ Hausdorff measure in $\mathbb{R}^{n+1}$ and set $Z=\{x\in X \mid e_{n+1}\perp T_xX \}$, where $T_xX$ is the approximate tangent and $e_{n+1}$ is the unit vector in the last coordinate.
It is clear from the co-area formula that one can write $Z=Z_1\cup Z_2$ where $\mathcal{H}^{k}(Z_1\cap \{x_{n+1}=t\})=0$ for every $t\in \mathbb{R}$ and the projection of $Z_2$ to the last co-ordinate, $P_{n+1}(Z_2)\subseteq \mathbb{R}$ is of Lebesgue measure zero.
Is it also clear that, in fact, $\mathcal{H}^{k+1}(Z_1)=0$?
(Trying to understand the proof of 11.6 in Ilmanen's "elliptic regularization" book).
