On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

Abstract

Let $Ω$ be a countable infinite product $Ω^\N$ of copies of the same probability space $Ω_1$, and let ${Ξ_n}$ be the sequence of the coordinate projection functions from $Ω$ to $Ω_1$. Let $Ψ$ be a possibly nonmeasurable function from $Ω_1$ to $\R$, and let $X_n(ω) = Ψ(Ξ_n(ω))$. Then we can think of ${X_n}$ as a sequence of independent but possibly nonmeasurable random variables on $Ω$. Let $S_n = X_1+...+X_n$. By the ordinary Strong Law of Large Numbers, we almost surely have $E_*[X_1] \le \liminf S_n/n \le \limsup S_n/n \le E^*[X_1]$, where $E_*$ and $E^*$ are the lower and upper expectations. We ask if anything more precise can be said about the limit points of $S_n/n$ in the non-trivial case where $E_*[X_1] < E^*[X_1]$, and obtain several negative answers. For instance, the set of points of $Ω$ where $S_n/n$ converges is maximally nonmeasurable: it has inner measure zero and outer measure one.