The Law of Large Numbers for independent identically distributed nonmeasurable random variables

Fact: For any real-valued function f, measurable or not, on a probability space, there exists a largest measurable function f^L such that f^L≤f and a smallest measurable function f^U such that f^U≥f, and f^L and f^U are unique up to almost sure equality.

Definition: A set U in a probability space is maximally nonmeasurable providing all its measurable subsets have measure zero and all its measurable supersets have measure one.

Definition: A sequence X₁,X₂,... of independent identically distributed not necessarily measurable random variables will be a sequence of functions on an infinite product of copies of a probability space, such that X_n(w₁,w₂,...)=F(w_n) for each n and a single fixed function F.

Henceforth suppose X₁,X₂,... are like that. Let S_n=X₁+...+X_n.

Easy consequence of the Law of Large Numbers: If X₁^L and X₁^U have finite expectations, then almost surely E[X₁^L]≤ liminf S_n/n≤ limsup S_n/n≤E[X₁^U].

Can one strengthen this? E.g., can one hope that one of the inequalities is an equality? Yesterday I finished proving a negative answer.

Theorem: Suppose X₁^L and X₁^U are integrable. Let A be any proper non-empty subset of the interval [E[X₁^L],E[X₁^U]] (which implies that E[X₁^L]<E[X₁^U]). Consider the respective subsets of our probability space where:

• lim S_n/n exists
• lim S_n/n exists and is in A
• limsup S_n/n is in A
• liminf S_n/n is in A
• all the limit points of S_n/n are in A

Then each of these subsets is maximally nonmeasurable.

This has a very interesting consequence for the philosophy of science, namely that unless we assume at the outset that what we are observing in the real world are measurable random variables, we can never come to that conclusion on the basis of observation of frequencies. For non-trivial cases (i.e., ones where E[X₁^L]<E[X₁^U]) of nonmeasurable random variables can equally well give neat limiting frequencies and not give them—any such limiting outcome is itself probabilistically maximally nonmeasurable.