1 Empirical Distributions

In this section, we let \(\mathcal{X}\) be a Polish space. The topology on \(\mathcal{X}\) will be generated by a metric \(d\), and we equip \(\mathcal{X}\) with its Borel \(\sigma \)-algebra \(\mathcal{B}(\mathcal{X})\). Further \((\Omega , \mathcal{F})\) will be a general measurable space (we do not fix a measure on this space), and \(n\) will denote an arbitrary natural number. We will also sometimes refer to a second Polish space \(\mathcal{Y}\) with associated metric \(\tilde d\), also equipped with its Borel \(\sigma \)-algebra. The space of measures on \(\mathcal{X}\) will be denoted \(\mathcal{M}(\mathcal{X})\), and the space of probability measures on \(\mathcal{X}\) will be denoted \(\mathcal{P}(\mathcal{X})\). We endow \(\mathcal{M}(\mathcal{X})\) with the \(\sigma \)-algebra generated by the projection maps \(\{ \pi _{A} : \mathcal{M}(\mathcal{X}) \to \mathbb {R}\} _{A \in \mathcal{B}(\mathcal{X})}\), and similarly for \(\mathcal{P}(\mathcal{X})\).

We will also consider the space of multisets of elements in \(\mathcal{X}\). This will be given the disjoint union topology, where the union is over the \(n\)th symmetric power \(\mathrm{Sym}^n \mathcal{X}\) for \(n \in \mathbb {N}\). The \(n\)th symmetric power has the quotient topology inherited from \(\mathcal{X}^n\)