  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.
