Sankhya A

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An Abstract Law of Large Numbers

  • Nabil I. Al-Najjar
  • Luciano PomattoEmail author


We study independent random variables (Zi)iI aggregated by integrating with respect to a nonatomic and finitely additive probability ν over the index set I. We analyze the behavior of the resulting random average \({\int }_I Z_i d\nu (i)\). We establish that any ν that guarantees the measurability of \({\int }_I Z_i d\nu (i)\) satisfies the following law of large numbers: for any collection (Zi)iI of uniformly bounded and independent random variables, almost surely the realized average \({\int }_I Z_i d\nu (i)\) equals the average expectation \({\int }_I E[Z_i]d\nu (i)\).


Finitely additive probabilities Measure theory Measurability 

AMS (2000) subject classification

Primary 28A25 Secondary 60F15 


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Copyright information

© Indian Statistical Institute 2019

Authors and Affiliations

  1. 1.Department of Managerial Economics and Decision Sciences, Kellogg School of ManagementNorthwestern UniversityEvanstonUSA
  2. 2.Division of the Humanities and Social SciencesCalifornia Institute of TechnologyPasadenaUSA

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