Approximations of expectations under infinite product measures

Ashkenazi-Golan, Galit ORCID: 0000-0003-3896-4131, Flesch, János, Predtetchinski, Arkadi and Solan, Eilon (2024) Approximations of expectations under infinite product measures. Journal of Mathematical Analysis and Applications. ISSN 0022-247X (In Press)

Text (Finitistic_approximations (1)) - Accepted Version Pending embargo until 1 January 2100. Available under License Creative Commons Attribution Non-commercial No Derivatives. Download (172kB)

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.

Item Type:	Article
Additional Information:	© 2025
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	07 Jan 2025 11:36
Last Modified:	07 Jan 2025 11:39
URI:	http://eprints.lse.ac.uk/id/eprint/126607

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