On games with incomplete information and the Dvoretsky-Wald-Wolfowitz theorem with countable partitions

It has remained an open question as to whether the results of Milgrom-Weber [Milgrom, P.R., Weber, R.J., 1985. Distributional strategies for games with incomplete information. Mathematics of Operations Research 10, 619-632] are valid for action sets with a countably infinite number of elements without additional assumptions on the abstract measure space of information. In this paper, we give an affirmative answer to this question as a consequence of an extension of a theorem of Dvoretzky, Wald and Wolfowitz (henceforth DWW) due to Edwards [Edwards, D.A., 1987. On a theorem of Dvoretsky, Wald and Wolfowitz concerning Liapunov measures. Glasgow Mathematical Journal 29, 205-220]. We also present a direct elementary proof of the DWW theorem and its extension, one that may have an independent interest.