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2013 | 61 | 2 | 161-168
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On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

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Let Ω be a countable infinite product $Ω₁^{ℕ}$ of copies of the same probability space Ω₁, and let {Ξₙ} be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of {Xₙ} as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have $E_{*}[X₁] ≤ lim inf Sₙ/n ≤ lim sup Sₙ/n ≤ E*[X₁]$, where $E_{*}$ and E* are the lower and upper expectations. We ask if anything more precise can be said about the limit points of Sₙ/n in the nontrivial case where $E_{*}[X₁] < E*[X₁]$, and obtain several negative answers. For instance, the set of points of Ω where Sₙ/n converges is maximally nonmeasurable: it has inner measure zero and outer measure one.
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  • Department of Philosophy, Baylor University, One Bear Place #97273, Waco, TX 76798-7273, U.S.A.
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