Let $S\subset \mathbb R^N$ be a $\mathcal H^{N-1}$-rectifiable set. Then we know that there exist countably many Lipschitz $N-1$-graphs $\Gamma_i\subset \mathbb R^N$ such that $$ \mathcal H^{N-1}\left(S\setminus \bigcup_{i=1}^\infty \Gamma_i\right)=0 $$ Without lose of generality, we assume that $$ \mathcal H^{N-1}(\Gamma_i\cap \Gamma_j)=0\text{ if }i\neq j $$ Hence, for $\mathcal H^{N-1}$ a.e. $x\in S$, there exists only one $\Gamma_i$ such that $x\in\Gamma_i$, and let us name this $\Gamma_i$ by $\Gamma_x$.
My question: would it possible to hold that $$ \lim_{r\to 0}\frac{\mathcal H^{N-1}(\Gamma_x\cap B(x,r))}{r^{N-1}}=1 $$ for $\mathcal H^{N-1}$ a.e. $x\in S$?
Thank you!