My question is about lebesgue density theorem:
Let $\mathcal{H}^s$ be $s-$dimensional Hausdorff measure.
If $A\subset \mathbb{R}^{n}$ with $0<\mathcal{H}^s(A)<\infty,$ then for $\mathcal{H}^{s}$ almost all $x\in A,$
$\limsup_{r \rightarrow 0}\frac{\mathcal{H}^{s}(A\cap B(x,r))}{\beta_s r^s}\leq 1,$ where $\beta_s$ is the $s-$dimensional Hausdorff measure of $s-$dimensional unit ball.
Do we have the above inequality for all $x\in A$, if we assume that $A$ is a subset of a $C^{1}-$manifold?
Thank you so much