Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(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 $M$

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