I'm reading papers about concentration of measure. Could you check if my below understanding is correct?
Let $(E, d)$ be a metric space and $\mu$ a probability measure on the Borel $\sigma$-algebra of $E$. Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $(X_n)_n$ with $X_n:(\Omega, \mathcal F, \mathbb P) \to E$ a sequence of i.i.d. random variables whose common distribution on $E$ is $\mu$. This implies $\mu$ is the push-forward of $\mathbb P$ by $X_n$ for all $n$.
We define a sequence of empirical measures on $E$ by $$ \mu_n := \sum_{i=1}^n \delta_{X_i} \quad \forall n \ge 1. $$
Let $\mathscr P(E)$ be the space of all Borel probability measures on $E$ and $d'$ a metric on $\mathscr P(E)$. Clearly, $\mu \in \mathscr P(E)$ and $\mu_n:(\Omega, \mathcal F, \mathbb P) \to \mathscr P(E)$ is a random variable for all $n$. Then we are interested in the convergence rate of $$ \mathbb E[d'(\mu_n, \mu)] $$ and the bounds of $$ \mathbb P[d'(\mu_n, \mu) - \mathbb E[d'(\mu_n, \mu)] \ge t]. $$
Here the expectation $\mathbb E$ is computed w.r.t. the probability measure $\mathbb P$.