Let $A\subset\mathbb{R}^n$ be a bounded semi-algebraic subset with Lebesgue measure $\mu^n(A)=0$, $\mu^{n-1}(A)>0$ on some $(n-1)$-dimensional Hyperplane $H$ ("bounded" added ss, Bernd). Let $B(\epsilon)$ be an open $\epsilon$-Ball around zero.

Obviously, for any $\epsilon>0$, $\mu^n(A+B(\epsilon))>\mu^n(B(\epsilon))>0$

It seems just as obvious that also the following holds:

$\forall \delta>0, \exists \epsilon>0$ with $\mu^n(A+B(\epsilon))<\delta \ \ \ \ \ $ (1)

Since it is easy to find $A$ not being semi-algebraic for which the statement doesn't hold, I have problems in capturing the "niceness" of A in a proof.

Questions:

- is (1) correct?
- if so, what is a necessary / sufficient condition of A for (1) to hold?

Your help is very much appreciated!