Individual Level Randomness in a Nonatomic Population

This paper provides a construction of an uncountable family of i.i.d. random vectors, indexed by the points of a nonatomic measure space, such that (a) samples are measurable functions from the index space, and (b) an exact analogue of the Glivenko-Cantelli theorem holds with respect to the measure on that space. That is, a sample possesses a.s. the same distribution as that of the random vectors from which it is drawn. Moreover, any subspace of the index space with positive measure inherits the same property. This homogeneity property is important for an application of the construction to mathematical economics.