发表机构
Department of Mathematics University of Innsbruck（因斯布鲁克大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过范畴论中的自然性概念，证明了乘积测度是唯一自然的构造方式，并在标准Borel空间下推广到相对乘积测度。
AI 中文摘要
我们证明，对于可测空间上的每对概率测度，乘积测度是赋予其乘积空间上概率测度的唯一自然方式。这里，自然性是在范畴论意义下理解的，等同于该赋值与沿可测映射的推前运算可交换的条件。在标准Borel设定下，我们还证明了在固定基概率空间上的相对乘积测度的类似结果。
英文摘要
We show that the formation of the product measure is the only natural way to assign to each tuple of probability measures on measurable spaces a probability measure on the product space. Here, naturality is meant in the sense of category theory and amounts to the condition that the assignment commutes with pushforward along measurable maps. In the standard Borel setting, we also prove an analogous result for relative products over a fixed base probability space. As a corollary, we conclude that the formation of relative products is a lax symmetric monoidal functor.
Comments14 pages, v2: extended the results to tuples of probability measures
