Definition:Lebesgue Measure

Lebesgue Measure on the Reals
For a given set $$S \subseteq \R$$, let $$\left\{{I_n}\right\}$$ be a countable set of open intervals such that


 * $$S \subseteq \bigcup I_n$$

For the set of all subsets $$\mathcal{P} \left({\R}\right)$$ of the reals $$\R$$, construct a function $$\mu^*:\mathcal{P} \left({\R}\right) \to \R_+$$ as


 * $$m^*(S) = \inf_{\left\{{I_n}\right\} :S \subseteq \cup I_n} \sum l (I_n) \ $$

where the infimum ranges over all such sets $$\left\{{I_n}\right\} \ $$, and $$l(I_n) \ $$ is the length of the interval.

Then $$\mu^*\ $$ is known as the Lebesgue outer measure and can be shown to be an outer measure.

When the domain of $$\mu^*\ $$ is restricted to the set $$\mathfrak {M} \ $$ of Lebesgue-measurable sets, $$\mu^*\ $$ is instead written as $$\mu\ $$ and is known as the Lebesgue measure. Moreover, $(\R, \mathfrak M, \mu)$ is a measure space.