On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables
Volume 61 / 2013
Abstract
Let $\varOmega$ be a countable infinite product $\varOmega_{1}^{\mathbb N}$ of copies of the same probability space $\varOmega_1$, and let $\{ \varXi_n \}$ be the sequence of the coordinate projection functions from $\varOmega$ to~$\varOmega_1$. Let $\varPsi$ be a possibly nonmeasurable function from $\varOmega_1$ to $\mathbb R$, and let $X_n(\omega) = \varPsi(\varXi_n(\omega))$. Then we can think of $\{ X_n \}$ as a sequence of independent but possibly nonmeasurable random variables on $\varOmega$. Let $S_n = X_1+\cdots+X_n$. 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 nontrivial case where $E_*[X_1] < E^*[X_1]$, and obtain several negative answers. For instance, the set of points of $\varOmega$ where $S_n/n$ converges is maximally nonmeasurable: it has inner measure zero and outer measure one.
