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arXiv · 2608.08581

Loeb Equivalence for General Internal Probability Spaces

Haosui Duanmu·Xinyu Liu·David Schrittesser
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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.

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Haosui Duanmu, Xinyu Liu, David Schrittesser. 2026-08-09. Loeb Equivalence for General Internal Probability Spaces. https://arxiv.org/abs/2608.08581

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