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Jordan Measure


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] iff 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.


This entry contributed by John Derwent

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References

Shenitzer, A. and Steprans, J. "The Evolution of Integration." Amer. Math. Monthly 101, 66-72, 1994.

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Jordan Measure

Cite this as:

Derwent, John. "Jordan Measure." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/JordanMeasure.html

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