Approximations of expectations under infinite product measures

Galit Ashkenazi-Golan, János Flesch, Arkadi Predtetchinski*, Eilon Solan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We are given a bounded Borel-measurable real-valued function on a product of countably many Polish spaces, and a product probability measure. We are interested in points in the product space that can be used to approximate the expected value of this function. We define two notions. A point is called a weak ϵ-approximation, where ϵ≥0, if the Dirac measure on this point, except in finitely many coordinates where another measure can be taken, gives an expected value that is ϵ-close to the original expected value. A point is called a strong ϵ-approximation if the same holds under the restriction that in those finitely many coordinates the measure is equal to the original one. We prove that both the set of weak 0-approximation points and the set of strong ϵ-approximation points, for any ϵ>0, have measure 1 under the original measure. Finally, we provide two applications: (i) in Game Theory on the minmax guarantee levels of the players in games with infinitely many players, and (ii) in Decision Theory on the set of feasible expected payoffs in infinite duration problems.

Original languageEnglish
Article number129173
JournalJournal of Mathematical Analysis and Applications
Volume546
Issue number1
DOIs
StatePublished - 1 Jun 2025

Keywords

  • Infinite product space
  • Minmax value
  • Product measure
  • Strategic-form game
  • Universally measurable set

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