  We propose a new, more general definition of extended probability measures.
We study their properties and provide a behavioral interpretation. We put them
to use in an inference procedure, whose environment is canonically represented
by the probability space $(\Omega,\mathcal{F},P)$, when both $P$ and the
composition of $\Omega$ are unknown. We develop an ex ante analysis -- taking
place before the statistical analysis requiring knowledge of $\Omega$ -- in
which the true composition of $\Omega$ is progressively learned. We describe
how to update extended probabilities in this setting, and introduce the concept
of lower extended probabilities. We apply our findings to a species sampling
problem and to the study of the boomerang effect (the empirical observation
that sometimes persuasion yields the opposite effect: the persuaded agent moves
their opinion away from the opinion of the persuading agent).
