There are many ways to construct the Lebesgue measure, and different definitions are available for a subset $A\subset\mathbb{R}$ to be 'measurable'. I believe the most common definition is to use Caratheodory extension theorem involving the concept of an outer-measure; Royden's book uses this definition.
Another way to construct the Lebesgue measure is to first define a positive linear functional on $C_c(\mathbb{R})$ using the Riemann integration, and then applying the Riesz representation theorem on locally compact Hausdorff spaces; Rudin's book uses this approach, but I personally think that this is using a sledgehammer to crack a nut. Yet another approach uses both the concepts of an outer-measure and an inner-measure. We first define a subset with a finite outer-measure to be 'measurable' if its outer-measure equals its inner-measure. Then for a general subset, we take intersections with 'measurable' subsets and define things the way they're supposed to be. This approach is taken in Frank Jones' book.
All these various ways make me wonder what Lebesgue's original construction was. How did he construct the measure? What was he definition of a 'measurable' subset of $\mathbb{R}$? I don't speak French, so it is very hard for me to go through his original dissertations, and unfortunately it seems that many modern books lack this historic information.
So, what was Lebesgue's definition?