Title: An Abstract Law of Large Numbers
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).