The Lebesgue measure is an extension of the classical notions of length and area to more complicated sets (Hunter 2011, pp. 17-18). On
open
set union of pairwise disjoint open intervals . Its Lebesgue measure is
Given a closed set 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
Explore with Wolfram|Alpha
References Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved
Problems in Geometry. 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. 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 https://mathworld.wolfram.com/LebesgueMeasure.html
Subject classifications
, let an open
set 
be written as a union of pairwise disjoint open intervals. Its Lebesgue measure is
![S^'=[a,b]\ union _k(a_k,b_k)](/images/equations/LebesgueMeasure/Inline3.svg)
, where the open
intervals are pairwise disjoint and contained
in 
,
