Existence of empirical sampling morphisms beyond standard Borel spaces

Construct empirical sampling morphisms for measurable spaces outside the class of standard Borel spaces, and identify structural conditions on such spaces that ensure the existence of empirical sampling morphisms satisfying permutation invariance and empirical adequacy.

Background

The paper constructs empirical sampling morphisms for all standard Borel spaces (finite sets, the natural numbers with the discrete σ-algebra, and the real line with the Borel σ-algebra), showing these satisfy permutation invariance and empirical adequacy.

The authors raise the question of whether similar constructions extend to other classes of measurable spaces, and what additional structure might be required to define empirical sampling morphisms outside the standard Borel setting.