In the textbook "Topics in Random Matrix Theory" by Tao (or his blog https://terrytao.wordpress.com/2010/02/02/254a-notes-4-the-semi-circular-law/), the (normalized) empirical spectral distribution (ESD) of Hermitian matrix $M_n$ is defined as: $$\mu_{\frac{1}{\sqrt{n}}M_n}:=\frac{1}{n}\sum_{i=1}^n\delta_{\lambda_j(M_n)/\sqrt{n}}\,,$$ and the expectation $\textbf{E}\mu_{\frac{1}{\sqrt{n}}M_n}$ is defined via Riesz representation theorem: $\int_{R}\phi\, d\textbf{E}\mu_{\frac{1}{\sqrt{n}}M_n} := \textbf{E}\int_{R}\phi\, d\mu_{\frac{1}{\sqrt{n}}M_n}$.
$\textbf{(1)}$ My first question is that why should we define the expectation this way? Is it equally valid if we define the expectation as: $$\textbf{E}\mu_{\frac{1}{\sqrt{n}}M_n} = \frac{1}{n}\sum_{i=1}^n\delta_{\,\textbf{E}\lambda_j(M_n)/\sqrt{n}}\,.$$ That is, the nonrandom probability measure which puts equal weights on the expectation of each eigenvalue of the random matrix $M_n$.
$\textbf{(2)}$ It is stated later that by using the following proposition (Proposition 14 in the link https://terrytao.wordpress.com/2010/01/09/254a-notes-3-the-operator-norm-of-a-random-matrix/):
For any $\lambda>0$,
$$\textbf{P}(|\|M\|_{op}-\textbf{E}\|M\|_{op}|\geq\lambda)\leq C\exp(-c\lambda^2)\,,$$
for some absolute constants $C,c>0$.
We see that $\textbf{E}\mu_{\frac{1}{\sqrt{n}}M_n}$ is uniformly sub-Gaussian:
$$\textbf{E}\mu_{\frac{1}{\sqrt{n}}M_n}\{|x|>\lambda\}\leq Ce^{-c\lambda^2n^2}\,.$$
My second question is how should we see this?