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 ∫ IZidν(i). We establish that any ν that guarantees the measurability of ∫ IZidν(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 ∫ IZidν(i) equals the average expectation ∫ IE[Zi] dν(i).
- Finitely additive probabilities
- Measure theory
- Primary 28A25
- Secondary 60F15
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty