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

$$\mu_L(S')\equiv(b-a)-\sum_k(b_k-a_k).$$

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

References

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.