2014/08/31 by Vitaly Vanchurin · 4 citations
Mathematics · Physics and Astronomy · #Borel measure #Countable set #Discrete mathematics #Lebesgue integration #Lebesgue measure #Mathematics #Measure (data warehouse) #Noncommutative and Quantum Gravity Theories #Probability measure #Pure mathematics #Quantum Mechanics and Applications #Sample space #Standard probability space #Uncountable set #advanced mathematical theories #gr-qc #hep-ph #hep-th #σ-finite measure
paper · pdf · doi:10.1103/physrevd.91.023511
published in Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields 91(2) (American Physical Society) · 12 pages
arxiv created 2014/12/29 · openalex publication_date 2015/01/15 · arxiv updated 2016/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We discuss eternal inflation in the context of classical probability spaces defined by a triplet: sample space, \ensuremathσ-algebra, and probability measure. We show that the measure problem is caused by the countable additivity axiom applied to the maximal \ensuremathσ-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.