A Real-Valued Measure on non-Archimedean Field Extensions of

We introduce a real-valued measure on non-Archimedean ordered fields that extend the field of real numbers . The definition of is inspired by the Loeb measures of hyperreal fields in the framework of Robinson’s analysis with infinitesimals. The real-valued measure turns out to be general enough to obtain a canonical measurable representative in for every Lebesgue measurable subset of , moreover the measure of the two sets is equal. In addition, 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 , the Levi-Civita field. In particular, we compare with the uniform non-Archimedean measure over 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 . Recall that this result is false for the current non-Archimedean integration over . The paper concludes with a discussion on the representation of the Dirac distribution by pointwise functions on non-Archimedean domains.