  The Levi-Civita field $\mathcal{R}$ is the smallest non-Archimidean ordered
field extension of the real numbers that is real closed and Cauchy complete in
the topology induced by the order. In an earlier paper [Shamseddine-Berz-2003],
a measure was defined on $\mathcal{R}$ in terms of the limit of the sums of the
lengths of inner and outer covers of a set by countable unions of intervals as
those inner and outer sums get closer together. That definition proved useful
in developing an integration theory over $\mathcal{R}$ in which the integral
satisfies many of the essential properties of the Lebesgue integral of real
analysis. Nevertheless, that measure theory lacks some intuitive results that
one would expect in any reasonable definition for a measure; for example, the
complement of a measurable set within another measurable set need not be
measurable.
  In this paper, we will give a characterization for the measurable sets
defined in [Shamseddine-Berz-2003]. Then we will introduce the notion of an
outer measure on $\mathcal{R}$ and show some key properties the outer measure
has. Finally, we will use the notion of outer measure to define a new measure
on $\mathcal{R}$ that proves to be a better generalization of the Lebesgue
measure from $\mathbb{R}$ to $\mathcal{R}$ and that leads to a family of
measurable sets in $\mathcal{R}$ that strictly contains the family of
measurable sets from [Shamseddine-Berz-2003], and for which most of the classic
results for Lebesgue measurable sets in $\mathbb{R}$ hold.
