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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

Certainly not. "Improve" $A$ by adding one extra point where the equality fails. You mention $s$-dimensional unit ball, so I guess $s$ is an integer? –  GEdgar Oct 13 '12 at 21:32

1 Answer 1

Answer depends on something that you left unsaid. If $s$ is an integer and the dimension of $C^1$ manifold is equal to $s$, then yes, the inequality holds for all $x\in A$.

If the dimension of manifold is greater than $s$, then it does not really help us in bounding the $s$-dimensional measure.


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