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An abstract law of large numbers. (English) Zbl 1440.60025

Summary: We study independent random variables \((Z_i)_{i \in I}\) aggregated by integrating with respect to a nonatomic and finitely additive probability \(\nu\) 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 \(\nu\) that guarantees the measurability of \(\int_I Z_i d\nu (i)\) satisfies the following law of large numbers: for any collection \((Z_i)_{i \in I}\) of uniformly bounded and independent random variables, almost surely the realized average \(\int_I Z_i d\nu (i)\) equals the average expectation \(\int_I E[Z_i] d \nu (i)\).

MSC:

60F15 Strong limit theorems
28A25 Integration with respect to measures and other set functions
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