Is there any coherent measure for eternal inflation?

TL;DR

This paper examines the challenge of defining a coherent measure for probabilities in eternal inflation, demonstrating that no measure satisfying certain axioms can be consistently formulated, highlighting unresolved foundational issues.

Contribution

The paper introduces axioms for measures in eternal inflation and proves the non-existence of such measures within a simple toy model, emphasizing fundamental limitations.

Findings

No measure obeying the axioms exists in the toy model

Some measures satisfy the axioms but are otherwise unacceptable

The problem of defining probabilities in eternal inflation remains unsolved

Abstract

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.