  An eternally inflating universe produces an infinite amount of spatial
volume, so every possible event happens an infinite number of times, and it is
impossible to define probabilities in terms of frequencies. This problem is
usually addressed by means of a measure, which regulates the infinities and
produces meaningful predictions. I argue that any measure should obey certain
general axioms, but then give a simple toy model in which one can prove that no
measure obeying the axioms exists. In certain cases of eternal inflation there
are measures that obey the axioms, but all such measures appear to be
unacceptable for other reasons. Thus the problem of defining sensible
probabilities in eternal inflation seems not be solved.
