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. * Department of Economics, University of Minnesota, and Research Department, Federal Reserve Bank of Minneapolis. (Internet address: [email protected]) A significant part of the research reported here was done during a visit to the Mathematical Sciences Research Institute in December, 1985. I would like to express my appreciation for the support and hospitality of the Institute. I would particularly like to thank William Zame for his comments and suggestions, and Jerome Keisler and Robert Anderson for their correspondence regarding an alternative approach via Loeb measure. This paper is available as ewp-ge/9402001 from the archive at econwpa.wustl.edu. It may be copied without fee, provided that copies are not made or distributed for direct commercial advantage and that credit to the source is given.