Abstract
In a previous paper [2], we developed a new Lebesgue-like measure on the Levi-Civita field \(\mathcal{R}\) that proved to be a strict improvement over the previously defined S-measure defined in [13, 9]. Nevertheless, we were only at first able to define such a measure for the one dimensional case leaving the case for higher dimensions as an open-ended question to be further researched. In another paper [15], the authors developed a generalization of the S-measure into higher dimensions using simplexes as their basic building blocks instead of boxes as simplexes proved to be more suitable for the topological structure of the Levi-Civita field \(\mathcal{R}\) . However, the resulting measure naturally inherited the same limitations that the original S-measure on \(\mathcal{R}\) had. In this new paper, we expand the same characterization given in [2] for the one-dimensional S-measurable sets to the S-measurable sets in \(\mathcal{R}^j\) as defined in [15] and develop our own generalization to higher dimensions for the measure given in [2].
