阁楼上的狮子--Borel-Kolmogorov悖论的一个解决方案
The lion in the attic -- A resolution of the Borel--Kolmogorov paradox
摘要：
关于先验概率为零的事件的条件作用的Borel-Kolmogorov悖论自从100多年前被发现以来一直吸引着学生和研究人员。经典条件作用只适用于正概率事件。如果我们忽略这种集合上的约束和条件，例如对于连续分布的随机变量$Y$的$\{Y=y\}$类型的事件，则几乎任何概率测度都可以被选择作为这种集合上的条件测量。对于这个悖论在概率为零的子集上的条件作用的出现，有许多描述和解释。然而，大多数治疗不提供明确的指示，如何避免它。我们建议通过定义一个版本的条件测量，利用豪斯多夫措施，以弥补这一差距。这使得选择规范的意义上，它只取决于空间的几何形状，从而消除任何歧义。我们描述了一组可能的措施中产生的背景下的Borel-Kolmogorov悖论和分类那些符合规范的措施。这份手稿的目的是提供一个手册的奇异条件概率：我们给出了明确的解释，其中设置歧义出现（和在哪里没有），以及如何摆脱这种歧义一劳永逸的典型选择。
英文摘要：
The Borel--Kolmogorov paradox of conditioning with respect to events of prior probability zero has fascinated students and researchers since its discovery more than 100 years ago. Classical conditioning is only valid with respect to events of positive probability. If we ignore this constraint and condition on such sets, for example events of type $\{Y=y\}$ for a continuously distributed random variable $Y$, almost any probability measure can be chosen as the conditional measure on such sets. There have been numerous descriptions and explanations of the paradox' appearance in the setting of conditioning on a subset of probability zero. However, most treatments don't supply explicit instructions on how to avoid it. We propose to close this gap by defining a version of conditional measure which utilizes the Hausdorff measure. This makes the choice canonical in the sense that it only depends on the geometry of the space, thus removing any ambiguity. We describe the set of possible measures arising in the context of the Borel--Kolmogorov paradox and classify those coinciding with the canonical measure. The objective of this manuscript is to provide a manual for singular conditional probability: We give an explicit explanation in which settings ambiguity arises (and where not) and how to get rid of this ambiguity once and for all by a canonical choice.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
