Hausdorff measure of an embedded submanifold

Question

Let $ M $ be a $ C^1 $- embedded n-submanifold (without boundary) of $ R^{n+k} $. Is it true that for every $ K $ compact set in $ R^{n+k} $ the n-dimensional Haussdorf measure of $ M \cap K $ is finite?

Thanks

Answer 1

Not in general, as the example $y=\sin(1/x)$, $0<x<1$, shows (given by Ilya as a comment).

But the answer is yes if $M$ is a closed subset of $\mathbb R^{n+k}$. Indeed, by the usual compactness argument it suffices to show that every point $x\in \mathbb R^{n+k}$ has a neighborhood $U$ such that $\mathcal H^n(M\cap U)<\infty$.

• If $x\notin M$ then there is a neighborhood $U$ such that $M\cap U=\varnothing$.
• If $x\in M$ then there is a neighborhood $V$ such that $M\cap V$ is diffeomorphic to a hyperplane with a diffeomorphism of $V$. This diffeomorphism is Lipschitz on any compactly contained neighborhood $U\Subset V$. This $U$ satisfies $\mathcal H^n(M\cap U)<\infty$.