Eternal Inflation, Global Time Cutoff Measures, and a Probability Paradox

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.