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-Löf tests allowed. The proof rests upon an application of Borel determinacy. Therefore, the proof 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 $\mathsf {ZFC}^-_n$. Here $\mathsf {ZFC}^-_n$ 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.

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中文翻译：

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连续测量的有效随机性

摘要：我们研究了哪些无限二进制序列（实数）相对于某些连续（即非原子）概率度量是有效随机的。我们证明，对于每个 $n$，除了可数的许多实数之外，所有的实数对于这种度量都是 $n$-随机的，其中 $n$ 表示允许的 Martin-Löf 检验的算术复杂性。证明依赖于 Borel 确定性的应用。因此，证明的前提是存在自然数幂集的无限次迭代。在论文的第二部分，我们提出了一个元数学分析，表明这个假设确实是必要的。更准确地说，存在一个可计算函数 $G$，使得对于任何 $n$，“关于连续概率测度的所有但可数的许多实数都是 $G(n)$-随机”的陈述不能在 $ \mathsf {ZFC}^-_n$。这里 $\mathsf {ZFC}^-_n$ 代表带有选择公理的 Zermelo-Fraenkel 集合论，其中幂集公理被 $n$ 的存在所取代——自然数幂集的多次迭代。后一个事实的证明依赖于对随机性的非常普遍的阻碍，即内部可定义性结构的存在。

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