Title: An Abstract Law of Large Numbers

Issue: Volume 82 Series A Part 1 Year 2020
Pages: 1 -- 12
We study independent random variables (Zi)i∈I 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 I Zidν(i). We establish that any ν that guarantees the measurability of I Zidν(i) satisfies the following law of large numbers: for any collection (Zi)i∈I of uniformly bounded and independent random variables, almost surely the realized average I Zidν(i) equals the average expectation I E[Zi]dν(i).
Primary 28A25; Secondary 60F15
Keywords and phrases: Finitely additive probabilities, Measure theory, Measurability