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Jordan Measure

  • Definition: Let M be a bounded set in the plane, i.e., M is contained entirely within a rectangle. The outer Jordan measure of M is the greatest lower bound of the areas of the coverings of M, consisting of finite unions of rectangles. The inner Jordan measure of M is the difference between the area of an enclosing rectangle S, and the outer measure of the complement of M in S. The Jordan measure, when it exists, is the common value of the outer and inner Jordan measures of M. If f is a bounded nonnegative function on the interval [a,b], the ordinate set of f is the set M={(x,y):x in [a,b],y in [0,f(x)]}. Then f is Riemann integrable on [a,b] if and only if M is Jordan measurable, in which case the Jordan measure of M is equal to int_a^bf(x)dx. There are analogous versions of Jordan measure in all other dimensions.

    Source: http://mathworld.wolfram.com

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