Nonstandard Measure Spaces with Values in non-Archimedean Fields

Abstract

The aim of this contribution is to bring together the areas of $p$-adic analysis and nonstandard analysis. We develop a nonstandard measure theory with values in a complete non-Archimedean valued field $K$, e.g. the $p-$adic numbers $\mathbb{Q}_p$. The corresponding theory for real-valued measures is well known by the work of P. A. Loeb, R. M. Anderson and others. We first review some of the standard facts on non-Archimedean measures and briefly sketch the prerequisites from nonstandard analysis. Then internal measures on rings and algebras with values in a nonstandard field ${^*K}$ are introduced. We explain how an internal measure induces a $K$-valued Loeb measure. The standard-part map between a Loeb space and the underlying standard measure space is measurable almost everywhere. We establish liftings from measurable functions to internal simple functions. Furthermore, we prove that standard measure spaces can be described as push-downs of hyperfinite internal measure spaces. This result is an analogue of a well-known Theorem on hyperfinite representations of Radon spaces. Then standard integrable functions are related to internal $S$-integrable functions and integrals are represented by hyperfinite sums. Finally, the results are applied to measures and integrals on $\mathbb{Z}_p$ and $\mathbb{Z}_p^{\times}$. We obtain explicit series expansions for the $p$-adic zeta function and the $p$-adic Euler-Mascheroni constant which we use for computations.