Visits to a nonmeasurable set and a new sceptical worry

Let X₁,X₂,... be a sequence of independent, identically distributed random variables. Let H be a set of values, and let R_n(H) be the proportion of X₁,...,X_n that are in H. Thus R_n(H)=V_n(H)/n, where V_n(H) is the number of times that the sequence X₁,...,X_n has visited H. We can call R_n(H) the rate of visits to H.

The strong Law of Large Numbers then shows that if H is a measurable set, then, almost surely (i.e., with probability one), R_n(H) converges to P(X₁ in H). We can use X₁ (or any of the other variables, since they are identically distributed) to induce a probability measure P₀ on the set of possible values via the formula P₀(H)=P(X₁ in H). Thus, for measurable H, almost surely, lim R_n(H)=P₀(H). I.e., the asymptotic rate of visits to a measurable set H is equal to the probability of that set.

But what if H is nonmeasurable? We could consider the general case, but let's simplify and make things more interesting. What if H is maximally nonmeasurable? A set H is maximally nonmeasurable with respect to a probability measure P₀ if and only if:

• All the measurable subsets of H have measure zero.
• All the measurable supersets of H have measure one.

On a reasonable assignment of interval-valued probabilities, a maximally nonmeasurable set is one that gets the full interval [0,1].

Such sets are intuitively a mess. So what should we expect the rate of visits to a maximally nonmeasurable set to behave like. It was my intuition that we can expect the rate of visits to be a mess—to not converge to any particular value.[note 1] Here's a precise way to formulate the question. Let B be any non-empty proper subset of the interval [0,1]. Form the following subsets of our original P-probability space:

• L(H): the set of points of the probability space such that lim R_n(H) exists.
• L_B(H): the set of points of the probability space such that lim R_n(H) exists and falls in B.
• I_B(H): the set of points of the probability space such that liminf R_n(H) falls in B.
• S_B(H): the set of points of the probability space such that limsup R_n(H) falls in B.

My intuition that we should expect the rate of visits to be a nonconvergent mess is an intuition that L(H) should have high probability, or, if it is itself nonmeasurable, it should contain a measurable subset of high probability. If some proofs that I haven't checked all the details of are correct, this intuition is wrong.

Conjecture (Theorem if my proofs are right): The sets L(H), I_B(H), S_B(H) and L_B(H) are all maximally nonmeasurable if H is maximally nonmeasurable.

If this is correct, then there is basically nothing probabilistic you can say about the asymptotic convergence of R_n(H) for a maximally nonmeasurable set H. You can't say that the rate of visits probably will converge (no surprise there) and you can't say that it probably won't.

So what? Well, consider now a new sceptical problem. We perform some experiment E a thousand times, and 405 times we get outcome H. We very reasonably want to conclude that the circumstances of the experiment E have approximately a 40% tendency of producing outcome H. And the greater the number of experiments we do, as long as the observed rate of H's is around 40%, the more confident we are of this judgment, with our confidence going to one in the limit.

But wait! What about the sceptical hypothesis that the objective chances are such that H is maximally nonmeasurable given E? It is tempting to say: "Well, that could be true, but the longer our sequence of experiments with a rate of around 40%, the more confident we should be that H is measurable and has measure around 40%. We just wouldn't expect to get such nice convergence if H were maximally nonmeasurable." However, the Conjecture, assuming it's correct, shows that as a piece of probabilistic reasoning, this is completely wrong. For it is neither likely nor unlikely on the maximal nonmeasurability hypothesis that we would observe an asymptotic rate of 40% (or of any other value). To see this, let B be the singleton {0.4}, and note that L_B(H) is maximally nonmeasurable. Thus, its interval-valued probability is all of [0,1], and we can have no probabilistic expectations about it.

If the maximal nonmeasurability hypothesis cannot be ruled out a posteriori, and yet must be ruled out, then it must be ruled out a priori. I think our best hope is a postulate that outcomes are always at least partly measurable, i.e., aren't maximally nonmeasurable. And that's a kind of Principle of Sufficient Reason.

I think my (unchecked) proofs of the Conjecture can generalize to give a more complicated result in the case of nonmaximally measurable sets.