What does the marginal probability density function of unordered eigenvalues refer to? For example, the semi-circle law
\begin{equation} f(x)=\frac{1}{2\pi}\sqrt{(4-x^2)^+} \end{equation} where $(x)^+=\text{max}\{0,x\}$
is the asymptotic marginal density of the unordered eigenvalues of a standard Wigner matrix under some conditions. Does this mean that $f(x)$ is the probability that any eigenvalue is equal to $x$?
No, since the distribution is continuous and is given by certain p.d.f.. The probability for each value is 0. To draw analogies with much easier setup: $\xi \sim \mathcal{U}[0, 1]$ with p.d.f. $f_{\xi}(x) = \mathbf{1}_x([0, 1])$, but the probability of uniform random variable being equal to $1/2$ is indeed not $1$.