A new sceptical argument

Let's say that an infinite sequence of real-numbered observations is generated by independent runs of a random process. Suppose that we can represent the runs of the random process as independent and identically distributed random variables X₁,X₂,.... Recall that a random variable is a function f from some probability space Ω to the reals R with the measurability property that f⁻¹[B] is a measurable subset of Ω for every Borel-measurable subset B of the reals R (and it's enough to check this for B an interval, since the intervals generate the Borel sets).

It turns out that under these assumptions we can almost surely recover the distribution of the random variables X_i from the observed sequence. For almost surely the frequency of the observations fitting into any given interval with rational numbers as ends will converge to the probability that X_i is in that interval. And since there are only countably many such intervals, almost surely for every such interval I we can read off the probability P(X_i in I) from the observed frequencies. And then by the uniqueness condition in the Caratheodory extension theorem, we can recover the probability of X_i being in A for any Borel subset, not just a rational-ended interval.

So far this sounds like a kind of vindication of infinitary frequentism. It is a helpful, optimistic result.

But notice a crucial assumption the recovery of the distribution of the X_i made: that the X_i are measurable when considered as functions to the Borel-measure space R. But there are infinitely many other σ-algebras on R besides the Borel one. When recovering the distribution from the observations, what justifies the assumption that the X_i are measurable as a function to R considered as coming with the Borel σ-algebra?

We might have some hope that the recovery process will be likely to fail if the X_i aren't measurable in this way, so that if the recovery process succeeds, we have good reason to think that the X_i have this measurability condition. But I suspect that some analogue of these results will show that this won't work.

So I guess we just need to assume measurability with respect to Borel sets. But why? Because God loves the Borel sets? It's not so crazy. I love the Borel sets, and God made me in his image, after all. :-)