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:


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

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.