Loeb Equivalence for General Internal Probability Spaces

Abstract

Loeb measure theory stands as one of the most influential concepts in nonstandard analysis, underpinning nearly all applications in probability, stochastic processes, and mathematical economics. The paper resolves a fundamental open problem in Loeb measure theory originally posed by Keisler and Sun: let $(Ω,\mathcal{F},μ)$ and $(Ω,\mathcal{G},ν)$ be two Loeb equivalent internal probability spaces, and $\mathcal H$ be the internal algebra generated from $\mathcal{F}\cup\mathcal{G}$. Does there exist an internal probability measure $P$ on $\mathcal H$ such that $(Ω,\mathcal{H},P)$ is Loeb equivalent to $(Ω,\mathcal{F},μ)$? While arXiv:2112.13955 recently provided a positive answer for hyperfinite probability spaces, the problem remained open for general internal probability spaces. We establish the existence of such an internal probability measure for all internal probability spaces.

Disclosure

“Gi which contradicts the choice of α. Thus indeed µ(A) ≤ η. So {Fout }i≤k is a tight cover. □ The next lemma on standard measure spaces was proved with the assistance of a large language model.”

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Drafting a complete proof for author revision

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2026-08-23T05:02:10.936007Z

Structural counts