I would like to calculate the limit value of a linear functional \begin{equation} \lim_{n\rightarrow\infty}\mathcal{I}_n=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n f(\lambda_i)=\lim_{n\rightarrow\infty}\int f(x)\mathrm{d}G_n(x), \end{equation} where $\lambda_i$ are eigenvalues of a $n\times n$ matrix and $G_n(x)$ is the empirical eigenvalue distribution function. In the large dimension limit, $G_n(x)$ converges to a nonrandom limit $G(x)$ but its explicit form is difficult to obtain. In stead, it is easy to obtain the Stieltjes transform of $G(x)$, say $m_G(x)$.

I hope I could obtain $\lim_{n\rightarrow\infty}\mathcal{I}_n$ using $m_G(x)$. Indeed, the equation (1.14) in "Z. D. Bai and J. W. Silverstein, CLT for Linear Spectral Statistics of Large-Dimensional Sample Covariance Matrices, Ann. Prob., 2004" shows \begin{equation} \int f(x)\mathrm{d}G(x)=-\frac{1}{2\pi i}\int f(z)m_G(z)\mathrm{d}z. \end{equation} But I don't understand how they obtain it and if it applies for general $G(x)$. I'd appreciate someone giving me some hints.