Let $p(\mathbf{x})$ be a non-zero polynomial in the polynomial ring $\mathbb{R}[\mathbf{x}]$. It is known that the affine variety $$ V = \{\mathbb{x} \in \mathbb{R}^{n}:p(\mathbf{x}) =0 \}$$ has Lebesgue measure zero in $\mathbb{R}^{n}$.
Now, consider the following set $$ S_{\varepsilon} = \{\mathbb{x} \in \mathbb{R}^{n}: |p(\mathbf{x})| \leq \varepsilon\},$$ with $\varepsilon > 0$.
1) What can be said about the Lebesgue measure of $S_{\varepsilon}$?
2) Let $\mu$ be a Lebesgue measure. Is it true that $\lim_{\varepsilon \rightarrow 0^{+}} \mu (S_{\varepsilon}) = \mu(V)=0$?
I just need some ideas to answer these questions. Thanks in advance!