AI 中文总结
该研究解决了勒贝测度理论中由基斯勒和孙提出的基础开放问题，证明了任意两个勒贝等价的内部概率空间生成的内部代数上存在满足条件的内部概率测度，填补了一般内部概率空间相关问题的空白。
AI 中文摘要
勒贝测度理论是非标准分析中最具影响力的概念之一，是概率、随机过程和数理经济学领域几乎所有应用的基础。本文解决了由基斯勒（Keisler）和孙（Sun）提出的勒贝测度理论中的一个基础开放问题：设$(\boldsymbol{\Omega},\mathcal{F},\mu)$和$(\boldsymbol{\Omega},\mathcal{G},\nu)$是两个勒贝等价的内部概率空间，$\boldsymbol{\mathcal{H}}$是由$\boldsymbol{\mathcal{F}}\cup\boldsymbol{\mathcal{G}}$生成的内部代数，是否存在$\boldsymbol{\mathcal{H}}$上的内部概率测度$\boldsymbol{P}$，使得$(\boldsymbol{\Omega},\mathcal{H},P)$与$(\boldsymbol{\Omega},\mathcal{F},\mu)$勒贝等价？尽管arXiv:2112.13955近期对超有限概率空间给出了肯定回答，但一般内部概率空间的该问题仍未解决。我们证明了所有内部概率空间都存在这样的内部概率测度。
英文摘要
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.
