(i) | $M = (\Xi, \mathcal{S})$ is a D1108: Measurable space |

(ii) | $X_1, \ldots, X_N : \Omega \to \Xi$ are each a D202: Random variable from $P$ to $M$ |

**empirical probability distribution measure**with respect to $X_1, \ldots, X_N$ is the D3650: Random probability measure \begin{equation} \Omega \times \mathcal{S} \to [0, 1], \quad (\omega, S) \mapsto \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \in S \}} (\omega) \end{equation}