This paper is a discussion note on Isaacs, Hájek and Hawthorne (2022), which claims to offer a new motivation for imprecise probabilities, based on the mathematical phenomenon of non-measurability. In this note, I clarify some consequences of that proposal. In particular, I show that if the proposal is applied to a bounded 3-dimensional space, then one has to reject at least one of the following: If A is at most as probable as B and B is at most as probable as C, then A is at most as probable as C. • Let A∩C=B∩C=∅. A is at most as probable as B if and only if (A∪C) is at most as probable as (B∪C). But rejecting either statement seems unattractive.

中文翻译：

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分配不可测集的不精确概率的后果


本文是对 Isaacs、Hájek 和 Hawthorne (2022) 的讨论笔记，该论文声称基于不可测量的数学现象为不精确概率提供了新的动机。在本说明中，我澄清了该提案的一些后果。特别是，我表明，如果该提案应用于有界 3 维空间，则必须拒绝至少以下之一：如果 A 的概率最多与 B 一样，并且 B 的概率最多与 C 一样，那么A 至多与C 一样可能。 • 令A∩C=B∩C=∅。 A 的概率最多与 B 一样当且仅当 (A∪C) 的概率最多与 (B∪C) 一样。但拒绝这两种说法似乎都没有吸引力。