New results on integration on the Levi-Civita field

Abstract

New results for integration of functions on the Levi-Civita field R are presented in this paper which is a continuation of the work done in Shamseddine and Berz (2003) [13] and complements it. For example, we show that if f and g are bounded on a measurable set A and f=g almost everywhere on A then f is measurable on A if and only if g is measurable on A in which case the integrals of f and g over A are equal. We also show that if A⊂R is measurable and if (fn) is a sequence of measurable functions that converge uniformly on A to f, then f itself is measurable on A and its integral over A is given by ∫Af=limn→∞∫Afn.