distribution according to random measures Is a random probability measure a probability measure, i.e. if $X_i$ is random variable on some probability space $(\Omega,\mathcal F,\mathbb P)$ could its induced distribution be a random probability measure $\mu$, say $X_i\sim \mu$?
Then, as $\mu$ is itself random, it has an own probability distribution, $\mu\sim Q$.
For example let's take the empirical measure $P_n(A)=\frac{1}{n}\sum_{i=1}^n \delta_{X_i}(A)$. Then the empirical distribution function $F_n(x):=\frac{1}{n}\sum_{i=1}^n 1\{X_i\le x\}$ is a random probability measure as it is an empirical measure indexed by the class $\mathcal C=\{(-\infty,x]: x\in \mathbb R\}$? Can one say $X_i\sim P_n$?
 A: For a random measure, it is just the case that $\Omega = \mathcal{P}$ the set of all probability measures on space. For example $P_n$ in your definition. 
The empirical measure is a random variable on the space of functions. This measure is obtained via taking the push forward measure of the mapping $\pi: \mathcal{P}(\mathbb{R}) \mapsto \mathcal{S}$, where $\mathcal{S}$ is the collection of step functions in $\mathbb{R}$.
$X\sim \mu$ is for a fixed measure $\mu$. In your example, you cannot say $X_i \sim P_n$ because the distribution $X_i$'s is something from which you have drawn the random variables $X_i$'s to construct the measure in the first place ($P_n$ is just a function of $X_i$'s). But, if you wanted mean that $X$ is a "sample" from the empirical measure, then you have to keep in mind that there are two steps of randomness. The first step to construct the measure, and the second is taking a sample from $P_n$ conditionally on the $X_i$'s in the first step. So one needs to write the distribution of $X$ using a conditional distribution.
