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Lebesgue Measure

An extension of the classical notions of length and Area to more complicated sets. Given an open set $S\equiv \sum_k
(a_k,b_k)$ containing Disjoint intervals,

\mu_L(S)\equiv \sum_k (b_k-a_k).

Given a Closed Set $S'\equiv [a,b] -\sum_k (a_k,b_k)$,


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


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.

© 1996-9 Eric W. Weisstein