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

Abstract

Let $\Omega$ be a countable infinite product ${\Omega }^{\mathbb{N}}$ 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 $\mathbb{R}$, 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 $E_{\ast }[X_1]\le \mathrm{liminf}{S}_n/n\le \mathrm{limsup}{S}_n/n\le E^{\ast }[X_1]$, where $E_{\ast }$ and $E^{\ast }$ 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 $E_{\ast }[X_1]<E^{\ast }[X_1]$, 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.