I am trying to learn mean field games from the Carmona Book, "Probabilistic Theory of Mean Field Games with Applications". There is a fundamental notion of empirical distribution.
The sate $X = (x_1, \cdots, x_n) \in E^n$ where $E$ is a compact metric space. The empirical distribution or probability measure $\overline{\mu}_X^n$ is defined as:
$\overline{\mu}_X^n = \frac{1}{n} \sum_{i = 1}^n \delta_{x_i}$ where the $\delta_x$ is a unit mass point at $x \in E$.
Other than compactness there is no assumption made on the state space. In what sense is $\overline{\mu}_X^n$ a measure on $E$.