  The definition of probabilities in eternally inflating universes requires a
measure to regulate the infinite spacetime volume, and much of the current
literature uses a global time cutoff for this purpose. Such measures have been
found to lead to paradoxical behavior, and recently Bousso, Freivogel,
Leichenauer, and Rosenhaus have argued that, under reasonable assumptions, the
only consistent interpretation for such measures is that time must end at the
cutoff. Here we argue that there is an alternative, consistent formulation of
such measures, in which time extends to infinity. Our formulation begins with a
mathematical model of the infinite multiverse, which can be constructed without
the use of a measure. Probabilities, which obey all the standard requirements
for a probability measure, can then be defined by mathematical limits. They
have a peculiar feature, however, which we call time-delay bias: if the outcome
of an experiment is reported with a time delay that depends on the outcome,
then the observation of the reports will be biased in favor of the shorter time
delay. We show how the paradoxes can be resolved in this interpretation of the
measure.
