An Abstract Law of Large Numbers

We study independent random variables (Z_i)_(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 ∫_IZ_idν(i). We establish that any ν that guarantees the measurability of ∫_IZ_idν(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 ∫_IZ_idν(i) equals the average expectation ∫_IE[Z_i]dν(i).