Empirical spectral measure ($L_N=N^{-1}{\sum }_{i=1}^N{\delta }_{{\lambda }_i}$)

For an $N$-by-$N$ Hermitian matrix, the empirical spectral measure puts mass $1/N$ at each eigenvalue, counted with multiplicity. It is a probability measure. Integrating a test function against it gives the average of that function evaluated at the eigenvalues.