Abstract

We prove that a Riemann-like integral on non-Archimedean extensions of \(\mathbb{R}\) cannot assign an integral to every function whose standard part is measurable and simultaneously satisfy the fundamental theorem of calculus. We also discuss how existing theories of non-Archimedean integration deal with the incompatibility of these conditions.

中文翻译：

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非阿基米德积分的不兼容结果

摘要

我们证明了\(\mathbb{R}\) 的非阿基米德扩展上的类黎曼积分不能为每个标准部分可测且同时满足微积分基本定理的函数分配积分。我们还讨论了现有的非阿基米德积分理论如何处理这些条件的不相容性。