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