Outer, Inner, and Lebesgue Measure are defined and systematically studied; first for (n-dimensional) intervals, then for finite and countable union of intervals, then for open and closed sets, and finally for general Lebesgue Measurable sets in Euclidean Spaces. The Approximation Theorem and the Caratheodory Characterization of Measurability are proven. Borel sets are studied and examples are given of Nonmeasurable Sets, as well as Measurable Sets which are not Borel.