Let be a bounded set in the plane, i.e., is contained entirely within a rectangle. The outer Jordan measure of is the greatest lower bound of the areas of the coverings of , consisting of finite unions of rectangles. The inner Jordan measure of is the difference between the area of an enclosing rectangle , and the outer measure of the complement of in . The Jordan measure, when it exists, is the common value of the outer and inner Jordan measures of .
If is a bounded nonnegative function on the interval , the ordinate set of f is the set
Then is Riemann integrable on iff is Jordan measurable, in which case the Jordan measure of is equal to .
There are analogous versions of Jordan measure in all other dimensions.