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

Abstract

Let $\Omega$ be a countable infinite product $\Omega^\N$ of copies of the same probability space $\Omega_1$, and let ${\Xi_n}$ be the sequence of the coordinate projection functions from $\Omega$ to $\Omega_1$. Let $\Psi$ be a possibly nonmeasurable function from $\Omega_1$ to $\R$, and let $X_n(\omega) = \Psi(\Xi_n(\omega))$. Then we can think of ${X_n}$ as a sequence of independent but possibly nonmeasurable random variables on $\Omega$. 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 $\Omega$ where $S_n/n$ converges is maximally nonmeasurable: it has inner measure zero and outer measure one.