The Lebesgue measure is an extension of the classical notions of length and area to more complicated sets (Hunter 2011, pp. 17-18). On , let an open set be written as a union of pairwise disjoint open intervals. Its Lebesgue measure is

Given a closed set , where the open intervals are pairwise disjoint and contained in ,

A unit line segment has Lebesgue measure 1; the Cantor set has Lebesgue measure 0. The Minkowski measure of a bounded, closed set is the same as its Lebesgue measure (Ko 1995).

See also

Banach-Tarski Paradox, Cantor Set, Complete Measure, Measure, Measure Universal Set, Riesz-Fischer Theorem Explore this topic in the MathWorld classroom

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References

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 4, 1991.Hunter, J. K. "Lebesgue Measure." Ch. 2 in Measure Theory. pp. 17-18, 2011. https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes.pdf.Kestelman, H. "Lebesgue Measure." Ch. 3 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 67-91, 1960.Ko, K.-I. "A Polynomial-Time Computable Curve Whose Interior Has a Nonrecursive Measure." Theoret. Comput. Sci. 145, 241-270, 1995.

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Lebesgue Measure

Cite this as:

Weisstein, Eric W. "Lebesgue Measure." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LebesgueMeasure.html

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