Measure Theory and Integration on the Levi-Civita Field

Khodr Shamseddine and Martin Berz
khodr@math.msu.edu, berz@msu.edu
Department of Mathematics and
Department of Physics and Astronomy
Michigan State University
East Lansing, MI 48824, USA

Abstract:

It is well known that the disconnectedness of a non-Archimedean totally ordered field in the order topology makes integration more difficult than in the real case. In this paper, we present a remedy to that difficulty and study measure theory and integration on the Levi-Civita field. After reviewing basic elements of calculus on the field, we introduce a measure that proves to be a natural generalization of the Lebesgue measure on the field of the real numbers. Then we define measurable functions on measurable sets and derive a simple classification of such functions. Finally, we show how to integrate measurable functions and we show that the resulting integral satisfies similar properties to those of the Lebesgue integral of real calculus.
AMS Subject Classification: 11S80, 12J25, 26E30, 46S10
Key Words: Levi-Civita field, non-Archimedean calculus, measurable sets, measurable functions, integration