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Lebesgue Measure

The Lebesgue measure is an extension of the classical notions of length and area to more complicated sets. Given an open set S=sum_(k)(a_k,b_k) containing disjoint intervals, the Lebesgue measure is defined by

mu_L(S)=sum_(k)(b_k-a_k).

Given a closed set S^'=[a,b]-sum_(k)(a_k,b_k),

mu_L(S^')=(b-a)-sum_(k)(b_k-a_k).

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

Cantor Set, Measure, 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.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 Web Resource. https://mathworld.wolfram.com/LebesgueMeasure.html

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