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 ![[a,b]](/images/equations/JordanMeasure/Inline11.svg)
, the ordinate set of f is the set
![M={(x,y):x in [a,b],y in [0,f(x)]}.](/images/equations/JordanMeasure/NumberedEquation1.svg) |
Then 
is Riemann integrable on ![[a,b]](/images/equations/JordanMeasure/Inline13.svg)
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.
