### Abstract

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 ∫
_{I}
Z
_{i}
dν(i). We establish that any ν that guarantees the measurability of ∫
_{I}
Z
_{i}
dν(i) satisfies the following law of large numbers: for any collection (Z
_{i}
)
_{i∈I}
of uniformly bounded and independent random variables, almost surely the realized average ∫
_{I}
Z
_{i}
dν(i) equals the average expectation ∫
_{I}
E[Z
_{i}
] dν(i).

Original language | English (US) |
---|---|

Journal | Sankhya A |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

### Fingerprint

### Keywords

- Finitely additive probabilities
- Measurability
- Measure theory
- Primary 28A25
- Secondary 60F15

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Sankhya A*. https://doi.org/10.1007/s13171-018-00162-z

}

*Sankhya A*. https://doi.org/10.1007/s13171-018-00162-z

**An Abstract Law of Large Numbers.** / Al-Najjar, Nabil; Pomatto, Luciano.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An Abstract Law of Large Numbers

AU - Al-Najjar, Nabil

AU - Pomatto, Luciano

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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 ∫ I Z i dν(i). We establish that any ν that guarantees the measurability of ∫ I Z i dν(i) satisfies the following law of large numbers: for any collection (Z i ) i∈I of uniformly bounded and independent random variables, almost surely the realized average ∫ I Z i dν(i) equals the average expectation ∫ I E[Z i ] dν(i).

AB - 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 ∫ I Z i dν(i). We establish that any ν that guarantees the measurability of ∫ I Z i dν(i) satisfies the following law of large numbers: for any collection (Z i ) i∈I of uniformly bounded and independent random variables, almost surely the realized average ∫ I Z i dν(i) equals the average expectation ∫ I E[Z i ] dν(i).

KW - Finitely additive probabilities

KW - Measurability

KW - Measure theory

KW - Primary 28A25

KW - Secondary 60F15

UR - http://www.scopus.com/inward/record.url?scp=85060199126&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85060199126&partnerID=8YFLogxK

U2 - 10.1007/s13171-018-00162-z

DO - 10.1007/s13171-018-00162-z

M3 - Article

AN - SCOPUS:85060199126

JO - Sankhya A

JF - Sankhya A

SN - 0976-836X

ER -