Measure Theory and Integration on the Levi-civita Field

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 and have similar properties. Then we introduce a family of simple functions from which we obtain a larger family of measurable functions and derive a simple characterization of such functions. We study the properties of measurable functions, we show how to integrate them over measurable sets of R, and we show that the resulting integral satisfies similar properties to those of the Lebesgue integral of real calculus.