What is an "empirical distribution"? Let $\lambda_{n,1} \leq \lambda_{n,2} \leq \dots \leq \lambda_{n,n} $ be the $n$ real eigenvalues of a random symmetric matrix, $X_n$ of dimension $n$. Then in this case one seems to define the "empirical distribution" as the ``random probability measure",
$L_n = \frac{1}{n} \sum_{i=1}^n \delta_{\lambda_{n,i}} $

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*Can someone kindly explain what exactly is this quantity?
I am used to thinking of probabilty measures as finitely additive functions on some $\sigma/Borel$ algebra of subsets of a set. This above thing $L_n$ doesn't look like anything like that!


*I don't understand why this $L_n$ seems capable of acting on functions like $<L_n,f>$. One typically wants to define the probability distribution $\bar{L}_n := \mathbb{E}[L_n] $ by the relation,  $<\bar{L}_n,f> := \mathbb{E} [ <L_n, f> ]$
Can someone kindly explain the above construction?


*Like a particular statement which makes sense is that if $f= x^k$ in the above then, $<\bar{L}_n, x^k > = \frac{1}{n} \mathbb{E} [ Tr [X_n^k ]]$
Can someone help derive this?
 A: Let $(\Omega, \mathcal A, \mathsf P)$ be a probability space, $(S,\mathcal S)$ a measurable space and $X \colon (\Omega, \mathcal A) \to (S, \mathcal S)$ a (general) random variable. Then, for $X = X(\omega)\in S$ one can define a measure on $S$ by setting for any $A \in \mathcal S$
$$ \delta_{X(\omega)} (A) = \begin{cases} 1, & \mbox{ if }X(\omega) \in A, \\ 0, & \mbox{ if }X(\omega)\notin A. \end{cases}$$
That is, the measure of $A$ is one if and only if the realization of $X$ is in $A$. Note that this measure is random because it depends on the value of $\omega$. Though, the argument of $\omega$ is usually suppressed in the notation.
For any function $f \colon S \to \mathbf R$ we can take integrals with respect to $\delta_X$
$$ \int_S f(x) \, d\, \delta_X(x) = f(X). $$
In other words, by taking integrals (expectations) of $f$ with respect to the measure $\delta_X$ we basically evaluate $f$ at the value of the random variable $X$. This follows from the definitions of $\delta_X$ - the Dirac measure at $X$, see also this post.
Similarly, for any random variables $X_1, \dots, X_n$ we can define the linear combination of the measures $\delta_{X_1}, \dots, \delta_{X_n}$ by setting for $A\in\mathcal S$
$$ \left(\frac{1}{n} \sum_{i=1}^n \delta_{X_i}\right) (A) := \frac{1}{n} \sum_{i=1}^n \delta_{X_i}(A).$$
$\frac{1}{n} \sum_{i=1}^n \delta_{X_i}$ is then called the empirical measure of the variables $X_1, \dots, X_n$, and is again a random measure on $(S,\mathcal S)$. We can take the expectation of $f$ with respect to the empirical measure
$$ \int_S f(x) \, d \,  \left(\frac{1}{n} \sum_{i=1}^n \delta_{X_i}\right)(x) = \frac{1}{n} \sum_{i=1}^n f(X_i).$$
If we now take the expectation of the previous formula (with respect to $X_1, \dots, X_n$), we get a real number
$$\mathbf E \frac{1}{n} \sum_{i=1}^n f(X_i) = \frac{1}{n} \sum_{i=1}^n \mathbf E f(X_i).$$
In your setting, $S = \mathbf R$ and $\lambda_{n,i}$, $i=1,\dots,n$ are the random variables used for the construction of the empirical measure $L_n$ as above. For $f \colon \mathbf R \to \mathbf R$ you have
$$ < L_n, f > = \int_\mathbf R f(x) \, d \, L_n(x) = \frac{1}{n} \sum_{i=1}^n f(\lambda_{n,i}),$$
which is a random variable, and 
$$ \mathbf E < L_n, f > = \frac{1}{n} \sum_{i=1}^n \mathbf E f(\lambda_{n,i}),$$
which is a real number. Therefore, $\overline L_n$ is a (non-random) measure on $\mathbf R$ such that for any $f$ 
$$ <\overline L_n,f> = \int_\mathbf R f(x) \, d \, \overline L_n (x) = \frac{1}{n} \sum_{i=1}^n \mathbf E f(\lambda_{n,i}). $$
In particular, for $f = x^k$ you get
$$ <\overline L_n,x^k> = \mathbf E < L_n, x^k > = \frac{1}{n} \sum_{i=1}^n \mathbf E \lambda_{n,i}^k,$$
where the last expression is equal to
$$ \frac{1}{n} \mathbf E\,\, Tr(X_n^k).$$
