Viewing the set of matrices $M_n (\mathbb{R})$ as embedded in $\mathbb{R}^{n \times n}$, I was wondering what was known about the Borel measure of standard matrix sets, such as $GL_n(\mathbb{R})$, $SL_n(\mathbb{R})$, $O_n$, etc. The same question also goes for $GL_n(\mathbb{C})$, and e.g. the unitary matrices $U_n$ and hermitian matrices.
Are these sets all measure zero?