The Continuum of Discrete Trajectories in Eternal Inflation

TL;DR

The paper analyzes the measure problem in eternal inflation, showing it arises from countable additivity issues and proposing a continuum-based solution using Lebesgue measure on the boundary of the inflationary multiverse.

Contribution

It demonstrates that the measure problem can be addressed by defining probability measures on a continuum of trajectories, resolving issues caused by countable additivity in infinite sample spaces.

Findings

The measure problem is linked to countable additivity in infinite spaces.

A continuum approach using Lebesgue measure can resolve the measure problem.

Some probabilistic questions remain unanswerable due to measure limitations.

Abstract

We discuss eternal inflation in context of classical probability spaces defined by a triplet: sample space, $\sigma$-algebra and probability measure. We show that the measure problem is caused by the countable additivity axiom applied to the maximal $\sigma$-algebra of countably infinite sample spaces. This is a serious problem if the bulk space-time is treated as a sample space which is thought to be effectively countably infinite due to local quantum uncertainties. However, in semiclassical description of eternal inflation the physical space expands exponentially which makes the sample space of infinite trajectories uncountable and the (future) boundary space effectively continuous. Then the measure problem can be solved by defining a probability measure on the continuum of trajectories or holographically on the future boundary. We argue that the probability measure which is invariant under the symmetries of the tree-like structure of eternal inflation can be generated from the Lebesgue measure on unit interval. According to Vitali theorem the Lebesgue measure leaves some sets without a measure which means that there are certain probabilistic questions in eternal inflation that cannot be answered.