Abstract

We introduce a real-valued measure \( m _L \) on non-Archimedean ordered fields \(( \mathbb{F} ,<)\) that extend the field of real numbers \(({\mathbb R},<)\). The definition of \( m _L \) is inspired by the Loeb measures of hyperreal fields in the framework of Robinson’s analysis with infinitesimals. The real-valued measure \( m _L \) turns out to be general enough to obtain a canonical measurable representative in \( \mathbb{F} \) for every Lebesgue measurable subset of \({\mathbb R}\), moreover the measure of the two sets is equal. In addition, \(m_L\) it is more expressive than a class of non-Archimedean uniform measures. We focus on the properties of the real-valued measure in the case where \( \mathbb{F} = \mathcal{R} \), the Levi-Civita field. In particular, we compare \( m _L \) with the uniform non-Archimedean measure over \( \mathcal{R} \) developed by Shamseddine and Berz, and we prove that the first is infinitesimally close to the second, whenever the latter is defined. We also define a real-valued integral for functions on the Levi-Civita field, and we prove that every real continuous function has an integrable representative in \( \mathcal{R} \). Recall that this result is false for the current non-Archimedean integration over \( \mathcal{R} \). The paper concludes with a discussion on the representation of the Dirac distribution by pointwise functions on non-Archimedean domains.

中文翻译：

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$$\mathbb{R}$$ 非阿基米德场扩展的实值测度

摘要

我们在扩展实数域\(({\mathbb R},<)的非阿基米德有序域\(( \mathbb{F} ,<)\)上引入实值测度\( m _L \ ) \)。\( m _L \)的定义受到罗宾逊无穷小分析框架中超真实场的 Loeb 测度的启发。实值测度\( m _L \)证明足够通用，可以在\( \mathbb{F} \)中为\({\mathbb R}\ )的每个 Lebesgue 可测子集获得一个规范的可测代表，此外两组的度量是相等的。此外，\(m_L\)它比一类非阿基米德统一测度更具表现力。我们关注实值测度在\( \mathbb{F} = \mathcal{R} \)的情况下的属性，即 Levi-Civita 场。特别是，我们将\( m _L \)与由 Shamseddine 和 Berz 开发的\( \mathcal{R} \ )上的统一非阿基米德测度进行比较，我们证明第一个与第二个非常接近，只要后者被定义为。我们还为 Levi-Civita 域上的函数定义了一个实值积分，并且我们证明了每个实连续函数在\( \mathcal{R} \)中都有一个可积的代表。回想一下，对于当前在\( \mathcal{R} \)上的非阿基米德积分，这个结果是错误的. 本文最后讨论了通过非阿基米德域上的点函数表示狄拉克分布。