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.
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Acknowledgments
Alessandro Berarducci and Mauro Di Nasso provided insightful comments to some ideas presented in Section 2.
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Bottazzi, E. A Real-Valued Measure on non-Archimedean Field Extensions of \(\mathbb{R}\). P-Adic Num Ultrametr Anal Appl 14, 14–43 (2022). https://doi.org/10.1134/S2070046622010022
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DOI: https://doi.org/10.1134/S2070046622010022