Thanks to visit codestin.com
Credit goes to mathworld.wolfram.com

TOPICS
Search

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

Explore with Wolfram|Alpha

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.

Referenced on Wolfram|Alpha

Lebesgue Measure

Cite this as:

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

Subject classifications