TY - JOUR
T1 - Approximations of expectations under infinite product measures
AU - Ashkenazi-Golan, Galit
AU - Flesch, János
AU - Predtetchinski, Arkadi
AU - Solan, Eilon
N1 - Publisher Copyright:
© 2024
PY - 2025/6/1
Y1 - 2025/6/1
N2 - 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.
AB - 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.
KW - Infinite product space
KW - Minmax value
KW - Product measure
KW - Strategic-form game
KW - Universally measurable set
UR - http://www.scopus.com/inward/record.url?scp=85214308855&partnerID=8YFLogxK
U2 - 10.1016/j.jmaa.2024.129173
DO - 10.1016/j.jmaa.2024.129173
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AN - SCOPUS:85214308855
SN - 0022-247X
VL - 546
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 1
M1 - 129173
ER -