Let $X_1, \ldots, X_N \in \text{Random}(\Omega \to \mathbb{R})$ each be a
D3161: Random real number such that
The
empirical probability distribution measure with respect to $X_1, \ldots, X_N$ is the
D3650: Random probability measure
\begin{equation}
\Omega \times \mathcal{B}(\mathbb{R}) \to [0, 1], \quad
(\omega, B) \mapsto \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \in B \}} (\omega)
\end{equation}