Effective Randomness for Continuous Measures

TL;DR

This paper explores the prevalence of effective randomness among infinite binary sequences with respect to continuous measures, establishing that almost all such sequences are n-random for each level of arithmetical complexity, under certain set-theoretic assumptions.

Contribution

It proves that for every arithmetical level n, all but countably many reals are n-random for some continuous measure, and shows the necessity of strong set-theoretic assumptions for this result.

Findings

Almost all reals are n-random for continuous measures at each complexity level.

The proof relies on Borel determinacy and iterated power sets, requiring strong set-theoretic assumptions.

There are limitations in proving these results within weaker set theories, due to internal definability structures.

Abstract

We investigate which infinite binary sequences (reals) are effectively random with respect to some continuous (i.e., non-atomic) probability measure. We prove that for every n, all but countably many reals are n-random for such a measure, where n indicates the arithmetical complexity of the Martin-L\"of tests allowed. The proof is based on a Borel determinacy argument and presupposes the existence of infinitely many iterates of the power set of the natural numbers. In the second part of the paper we present a metamathematical analysis showing that this assumption is indeed necessary. More precisely, there exists a computable function G such that, for any n, the statement `All but countably many reals are G(n)-random with respect to a continuous probability measure' cannot be proved in $ZFC^-_n$. Here $ZFC^-_n$ stands for Zermelo-Fraenkel set theory with the Axiom of Choice, where the Power Set Axiom is replaced by the existence of n-many iterates of the power set of the natural numbers. The proof of the latter fact rests on a very general obstruction to randomness, namely the presence of an internal definability structure.