The Levi-Civita field $\mathcal{R}$ is the smallest non-Archimedean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In this talk, I will define a new measure on $\mathcal{R}$ that leads to a family of measurable sets in $\mathcal{R}$ for which most of the classic results for Lebesgue measurable sets in $\mathbb{R}$ hold. Then I will generalize that measure to higher dimensions using simplexes as the basic building blocks for measurable sets. Then I will introduce a family of measurable functions defined on measurable subsets of $\mathcal{R}$ (resp. $\mathcal{R}^2$ or $\mathcal{R}^3$). I will study the properties of measurable functions, show how to integrate them over measurable subsets of $\mathcal{R}$ (resp. $\mathcal{R}^2$ or $\mathcal{R}^3$), and show that the resulting integral satisfies similar properties to those of the Lebesgue integral of Real Analysis.