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

TL;DR

This paper investigates the behavior of the average of a sequence of independent, possibly nonmeasurable random variables, revealing that the set where their averages converge is maximally nonmeasurable, with implications for the law of large numbers.

Contribution

It extends the classical law of large numbers to nonmeasurable random variables and characterizes the non-measurability of the convergence set.

Findings

The set of convergence points is maximally nonmeasurable.

Convergence of averages cannot be more precisely characterized in the nonmeasurable case.

The results highlight limitations of classical probabilistic laws for nonmeasurable 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.