  Problems with uniform probabilities on an infinite support show up in
contemporary cosmology. This paper focuses on the context of inflation theory,
where it complicates the assignment of a probability measure over pocket
universes. The measure problem in cosmology, whereby it seems impossible to
pick out a uniquely well-motivated measure, is associated with a paradox that
occurs in standard probability theory and crucially involves uniformity on an
infinite sample space. This problem has been discussed by physicists, albeit
without reference to earlier work on this topic. The aim of this article is
both to introduce philosophers of probability to these recent discussions in
cosmology and to familiarize physicists and philosophers working on cosmology
with relevant foundational work by Kolmogorov, de Finetti, Jaynes, and other
probabilists. As such, the main goal is not to solve the measure problem, but
to clarify the exact origin of some of the current obstacles. The analysis of
the assumptions going into the paradox indicates that there exist multiple ways
of dealing consistently with uniform probabilities on infinite sample spaces.
Taking a pluralist stance towards the mathematical methods used in cosmology
shows there is some room for progress with assigning probabilities in
cosmological theories.
