In Bayesian statistics, improper distributions and finitely additive probabilities (FAPs) are the two main alternatives to proper distributions, i.e. countably additive probabilities. Both of them can be seen as limits of proper distribution sequences w.r.t. to some specific convergence modes. Therefore, some authors attempt to link these two notions by this means, partly using heuristic arguments. The aim of the paper is to compare these two kinds of limits. We show that improper distributions and FAPs represent two distinct characteristics of a sequence of proper distributions and therefore, surprisingly, cannot be connected by the mean of proper distribution sequences. More specifically, for a sequence of proper distribution which converge to both an improper distribution and a set of FAPs, we show that another sequence of proper distributions can be constructed having the same FAP limits and converging to any given improper distribution. This result can be mainly explained by the fact that improper distributions describe the behavior of the sequence inside the domain after rescaling, whereas FAP limits describe how the mass concentrates on the boundary of the domain. We illustrate our results with several examples and we show the difficulty to define properly a uniform FAP distribution on the natural numbers as an equivalent of the improper flat prior. MSC 2010 subject classifications: Primary 62F15; secondary 62E17,60B10.

翻译：在贝叶斯统计学中，不适当分布和有限可加概率（FAPs）是适当分布（即可数可加概率）的两种主要替代方案。二者均可视为适当分布序列在特定收敛模式下的极限。因此，部分学者试图通过此途径将这两个概念联系起来，其中涉及部分启发式论证。本文旨在比较这两类极限。我们证明，不适当分布和FAPs体现了一个适当分布序列的两个不同特征，因此令人惊讶地，它们无法通过适当分布序列这一中间手段关联。具体而言，对于同时收敛到某个不适当分布和一组FAPs的适当分布序列，我们证明可以构造另一个具有相同FAP极限的适当分布序列，使其收敛到任意给定的不适当分布。该结果主要源于以下事实：不适当分布描述序列在域内重缩放后的行为，而FAP极限则描述质量在域边界上的集中方式。我们通过多个例子阐释结果，并展示在自然数集上正确定义均匀FAP分布作为不适当平坦先验等价物的困难性。MSC 2010主题分类：主类62F15；次类62E17，60B10。