the joint law of entries of a Guassian orthogonal ensemble matrix $G O E_n$ is given by $$ \mathbb{P}\left(\sqrt{n} \mathbf{X}_n \in B\right)=2^{-n / 2} \pi^{-n(n+1) / 4} \int_B e^{-\frac{1}{2} \operatorname{Tr}\left(\mathbf{X}^2\right)} d \mathbf{X} $$ for all Borel subsets $B$ of the space of $n \times n$ symmetric real matrices. Similarly, for a $G U E_n$ matrix $\mathbf{Z}_n$, $$ \mathbb{P}\left(\sqrt{n} \mathbf{Z}_n \in B\right)=2^{-n / 2} \pi^{-n^2 / 2} \int_B e^{-\frac{1}{2} \operatorname{Tr}\left(\mathbf{Z}^2\right)} d \mathbf{Z} $$ for all Borel subsets $B$ of the space of $n \times n$ Hermitian complex matrices. I don't understand the following part, why eigenvalues are almost surely distinct follows from above part.
Notice that the joint distribution of entries of $\mathbf{X}_n$ has a smooth density. Since the eigenvalues are continuous functions of the entries, it follows immediately that the eigenvalues of $\mathbf{X}_n$ are almost surely all distinct. (The event that two eigenvalues coincide has codimension $\geq 1$ in the $n$-dimensional variety of eigenvalues; the existence of a density thus means this set is null.)